JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 10, 2008 0651098283

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Volume 10,Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

January 2008

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL A quarterly international publication of Eudoxus Press, LLC Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles.Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See at the end instructions for preparation and submission of articles to JoCAAA. Webmaster:Ray Clapsadle Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http//:www.eudoxuspress.com.Annual Subscription Prices:For USA and Canada,Institutional:Print $277,Electronic $240,Print and Electronic $332.Individual:Print $87,Electronic $70,Print &Electronic $110.For any other part of the world add $25 more to the above prices for Print.No credit card payments. Copyright©2008 by Eudoxus Press,LLCAll rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.1,7-15,2008, COPYRIGHT 2008 EUDOXUS PRESS, 7 LLC

ON INVARIANT APPROXIMATION FOR NONCOMMUTATIVE MAPPINGS IN LOCALLY CONVEX SPACES

M. S. Khan Department of Mathematics and Statistics College of Science Sultan Qaboos University, P. O. Box 36, PCode 123, Al -Khod, Muscat, Sultanate of Oman. e-mail: [email protected] Hemant Kumar Nashine Department of Mathematics, Raipur Institute of Technology Chhatauna, Mandir Hasaud, Raipur-492101(Chhattisgarh), INDIA. e-mail: hemantnashine@redif f mail.com nashine 09@redif f mail.com ABSTRACT The aim of this paper is to generalize the results related to invariant approximation by weakening commutativity hypothesis and by increasing the number of mappings involved. 2000 Mathematics Subject Classification. 41A50, 46A03, 47H10. Key Words and Phrases: Best approximant, common fixed points, commutating mappings, compatible mapping, demiclosed mapping, locally convex space. 1. INTRODUCTION During the last four decades several interesting and valuable results as an application of fixed point theorems were studied extensively in the field of an invariant approximation theory. An excellent reference can be seen in [16]. Meinardus [9] was the first who employed a fixed-point theorem to establish the existence of an invariant approximation. Further, Brosowski [1] obtained a celebrated result and generalized the Meinardus’s result. Afterwards, a number of results exist has been proved in the direction of Brosowski [1] (see in [3, 12, 13, 17]). In a paper, Jungck and Sessa [5] further weakened the hypothesis of Sahab, Khan and Sessa [12] by replacing the weak and strong topology for relatively nonexpansive commutative maps. Recently, Nashine [10] obtained invariant approximation results for a class of contraction commutative three mappings in locally convex space and he extended all the previous related results. Some works on best approximation for weakly commutative mappings in locally convex spaces are done by Khan and Hussain [7, 11]. 1

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M. S. KHAN AND HEMANT KUMAR NASHINE

In [4] the concept of compatible mappings was introduced as a generalization of commuting mappings. The purpose of this paper is to further emulate the compatible mapping concept. We extend the result of Nashine [10] by employing compatible mappings in lieu of commuting mappings, and by using four mappings as opposed to three in the setup of in locally convex space. To achieve our goal, we use the concept given by K¨othe [8], Tarafdar [18] and the result of Jungck [5]. In this way, we give new direction to the line of investigation given by Brosowski [1]. Finally, we derive some consequences from our main result. 2. PRELIMINARIES Before we prove our main result, let us recall following definitions: Definition 2.1. [8, 7, 11]. In the sequel (E, τ ) will be a Hausdorff locally convex topological vector space. A family {pα : α ∈ I} of seminorms defined on E is said to be an Tnassociated family of seminorms for τ if the family {γU : γ > 0}, where U = i=1 Uαi and Uαi = {x : pαi (x) < 1}, forms a base of neighbourhood of zero for τ . A family {pα : α ∈ I} of seminorms defined on E is called an augmented associated family for τ if {pα : α ∈ I} is an associated family with the property that the seminorm max{pα , pβ } ∈ {pα : α ∈ I} for any α, β ∈ I. The associated and augmented families of seminorms will be denoted by A(τ ) and A∗ (τ ), respectively. It is well known that if given a locally convex space (E, τ ), there always exists a family {pα : α ∈ I} of seminorms defined of E such that {pα : α ∈ I} = A∗ (τ ). A subset M of E is τ -bounded if and only if each pα is bounded on M. The following construction will be crucial. Suppose that M is a τ -bounded subset of E. For this set M, we can select a number λα > 0 for each Tα∈I such that M ⊂ λα Uα where Uα = {x : pα (x) ≤ 1}. Clearly, B = α λα Uα is τ - bounded, τS -closed, absolutely convex and contains M. The linear span ∞ EB of B in E is n=1 nB. The Minkowski functional of B is a norm k.kB on EB . Thus, (EB , k.kB ) is a normed space with B as its closed unit ball and supα pα (x/λα ) = kxkB for each x ∈ EB . Definition 2.2. Let I and T be selfmaps on M. The map T is called (i) A∗ (τ )-nonexpansive if for all x, y ∈ M pα (T x − T y) ≤ pα (x − y), ∗

for each pα ∈ A (τ ). (ii)A∗ (τ )-I−nonexpansive if for all x, y ∈ M pα (T x − T y) ≤ pα (Ix − Iy), for each pα ∈ A∗ (τ ). For simplicity, we shall call A∗ (τ )-nonexpansive (A∗ (τ ) − I- nonexpansive) maps to be nonexpansive (I−nonexpansive). Following the concept of compatible due to Jungck [4], we have

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ON INVARIANT APPROXIMATION FOR NONCOMMUTATIVE MAPPINGS.....

3

Definition 2.3. [4]. A pair of self-mappings (T , I) of a locally convex space (E, τ ) is said to be compatible, if pα (T Ixn − IT xn ) → 0, whenever {xn } is a sequence in E such that T xn , Ixn → t ∈ E. Every commuting pair of mappings is compatible but the converse is not true in general. Definition 2.4. [10]. Let x0 ∈ M. We denote by PM (x0 ) the set of best M− approximant to x0 , i.e., if PM (x0 ) = {y ∈ M : pα (y − x0 ) = dpα (x0 , M) for all pα ∈ A∗ (τ )}, where dpα (x0 , M) = inf{pα (x0 − z) : z ∈ M}.

Definition 2.5. [10]. The map T : M → E is said to be demiclosed at 0 if for every net {xn } in M converging weakly to x and {T xn } converging strongly to 0, we have T x = 0. Throughout this paper F(T )(resp.F(I)) denotes the set of fixed point of mapping T (resp.I). The following result of Jungck [5] is needed in the sequel: Theorem 2.6. [5]. Let A, S, I and J be continuous self mappings of a compact metric space (X , d) with A(X ) ⊆ J (X ) and S(X ) ⊆ I(X ). If (A, I) and (S, J ) are compatible pairs and satisfying 1 d(Ax, Sy) < max{d(Ix, J y), d(Ix, Ax), d(J y, Sy), [d(Ix, Sy)+d(J y, Ax)]} > 0, 2 then A, S, I and J have a unique common fixed point in X . 3. MAIN RESULT Lemma 3.1. Let A and I be compatible self-maps of a τ -bounded subset M of a Hausdorff locally convex space (E, τ ). Then A and I be two compatible on M with respect to k.kB . Proof. By hypothesis for each pα ∈ A∗ (τ ), (3.1)

pα (AIxn − IAxn ) → 0,

whenever {xn } is a sequence in M such that Axn , Ixn → t ∈ M. Taking supremum on both sides, we get AIxn − IAxn sup pα ( )→0 λα α kAIxn − IAxn kB → 0 whenever {xn } is a sequence in M such that Axn , Ixn → t ∈ M.



We use a technique of Tarafdar [18] to obtain the following common fixed point theorems which generalize Theorem 2.6.

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M. S. KHAN AND HEMANT KUMAR NASHINE

Theorem 3.2. Let M be a nonempty τ -bounded, τ -sequentially compact subset of a Hausdorff locally convex space (E, τ ). Let A, S, I and J be self mappings of M with A(M) ⊆ J (M) and S(M) ⊆ I(M). If (A, I) and (S, J ) are compatible pairs, A and S are continuous, I and J are nonexpansive, and satisfying (3.2)

pα (Ax − Sy) < N(x, y)

where N(x, y) = max{pα (Ix − J y), pα (Ix − Ax), pα (J y − Sy), 1 2 [pα (Ix

− Sy) + pα (J y − Ax)]}

for all x, y ∈ M and pα ∈ A∗ (τ ), then A, S, I and J have a unique common fixed point in M. Proof. Since the norm topology on EB has a base of neighbourhood of zero consisting of τ -closed sets and M is τ -sequentially compact, therefore, M is a k.kB -sequentially compact subset of (EB , k.kB ) (Theorem 1.2, [18]). By Lemma 3.1, A and I are k.kB -compatible maps of M. Similarly, by the Lemma 3.1, S and J are k.kB -compatible maps of M. From (3.2) we obtain for x, y ∈ M, Ix−J y J y−Sy Ix−Ax supα pα ( Ax−Sy λα ) < max{supα pα ( λα ), supα pα ( λα ), supα pα ( λα ), 1 2 [supα

J y−Ax pα ( Ix−Sy )]}. λα ) + supα pα ( λα

Thus kAx − SykB < max{kIx − J ykB , kIx − AxkB , kJ y − SykB , (3.3) 1 2[

kIx − SykB + kJ y − AxkB ]}.

Note that, if I and J are nonexpansive on a τ -bounded, τ -sequentially compact subset M of E, then I and J are also nonexpansive with respect to k.kB and hence k.kB -continuous ([8]). A comparison of our hypothesis with that of Theorem 2.6 tells that we can apply Theorem 2.6 to M as a subset of (EB , k.kB ) to conclude that there exists a unique v ∈ M such that v = Av = Sv = Iv = J v. 

Theorem 3.3. Let M be a nonempty τ -bounded, τ -sequentially complete and p−starshaped subset of a Hausdorff locally convex space (E, τ ). Let A, S, I and J be self mappings of M with A(M) ⊆ J (M) and S(M) ⊆ I(M). Suppose (A, I) and (S, J ) are compatible pairs, A and S are continuous, I and J are nonexpansive, and affine, I(M) = M = J (M), p ∈ F(I) ∩ F(J ). If A, S, I and J satisfy the following: (3.4)

pα (Ax − Sy) < N(x, y)

where N(x, y) = max{pα (Ix − J y), pα (Ix − Ax), pα (J y − Sy), 1 2 [pα (Ix

− Sy) + pα (J y − Ax)]}

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ON INVARIANT APPROXIMATION FOR NONCOMMUTATIVE MAPPINGS.....

5

for all x, y ∈ M and pα ∈ A∗ (τ ), then A, S, I and J have a common fixed point in M provided one of the following conditions hold: (i) M is τ -sequentially compact; (ii) M is weakly compact in (E, τ ), I and J are weakly continuous and I − A and J − S are demiclosed at 0. Proof. Choose a monotonically nondecreasing sequence {kn } of real numbers such that 0 < kn < 1 and lim sup kn = 1. For each n ∈ N, define An , Sn : M → M as follows: (3.5)

An x = kn Ax + (1 − kn )p,

Sn x = kn Sx + (1 − kn )p.

Obviously, for each n, An and Sn map M into itself since M is p−starshaped. As I is affine, (A, I) is compatible and p ∈ F(I), so An Ix = kn AIx + (1 − kn )p IAn xn = I(kn Ax + (1 − kn )p = kn IAx + (1 − kn )Ip. Since (A, I) is compatible, therefore 0 ≤ limn pα (Am Ixn − IAm xn ) ≤ limn pα (AIxn − IAxn ) + limn (1 − km )pα (p − Ip) = 0, whenever limn In xn = limn Axn = t ∈ M for all n, for each x ∈ M. Hence {An } and I are compatible for each n and An (M) ⊆ M = J (M). Similarly, we can prove Sn and J are compatible for each n and Sn (M) ⊆ M = I(M). For all x, y ∈ M, pα ∈ A∗ (τ ) and for all j ≥ n, (n fixed), we obtain from (3.4) and (3.10) that pα (An x − Sn y) = kn pα (Ax − Sy) ≤ kj pα (Ax − Sy) ≤ pα (Ax − Sy) < max{pα (Ix − J y), pα (Ix − Ax), pα (J y − Sy), 1 2 [pα (Ix

− Sy) + pα (J y − Ax)]}

< max{pα (Ix − J y), pα (Ix − An x) + pα (An x − Ax), pα (J y − Sn y) + pα (Sn y − Sy), 1 2 [pα (Ix

− Sn y) + pα (Sn y − Sy)+

pα (J y − An x) + pα (An x − Ax)]}

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M. S. KHAN AND HEMANT KUMAR NASHINE

< max{pα (Ix − J y), pα (Ix − An x) +(1 − kn )pα (p − Ax), pα (J y − Sn y) +(1 − kn )pα (p − J y), 12 [pα (Ix − Sn y) +(1 − kn )pα (p − Sy) + pα (J y − An x) +(1 − kn )pα (p − Ax)]}. Hence for all j ≥ n, we have (3.6) pα (An x − Sn y) < max{pα (Ix − J y), pα (Ix − An x) +(1 − kj )pα (p − Ax), pα (J y − Sn y) +(1 − kj )pα (p − Sy), 12 [pα (Ix − Sn y) +(1 − kj )pα (p − Sy) + pα (J y − An x) +(1 − kj )pα (p − Ax)]}. As lim kj = 1, from (3.6), for every n ∈ N, we have (3.7) pα (An x − Sn y) = limj pα (An x − Sn y) < limj {max{pα (Ix − J y), pα (Ix − An x) +(1 − kj )pα (p − Ax), pα (J y − Sn y) +(1 − kj )pα (p − Sy), 12 [pα (Ix − Sn y) +(1 − kj )pα (p − Sy) + pα (J y − An x) +(1 − kj )pα (p − Ax)]}. This implies that for every n ∈ N, (3.8) pα (An x − Sn y) < max{pα (Ix − J y), pα (Ix − An x), pα (J y − Sn y), 1 2 [pα (Ix

− Sn y) + pα (J y − An x)]},

for all x, y ∈ M and for all pα ∈ A∗ (τ ). Moreover, I and J being nonexpansive on M, implies that I and J are k.kB -nonexpansive and, hence, k.kB -continuous. Since the norm topology on EB has a base of neighbourhood of zero consisting of τ -closed sets and M is τ -sequentially complete, therefore, M is a k.kB -sequentially complete subset of

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ON INVARIANT APPROXIMATION FOR NONCOMMUTATIVE MAPPINGS.....

7

(EB , k.kB ) (see proof of Theorem 1.2 in [18]). Thus from Theorem 3.2 with the condition (a) or (b), for every n ∈ N, An , Sn , I and J have unique common fixed point xn in M, i.e., (3.9)

xn = An xn = Sn xn = Ixn = J xn ,

for each n ∈ N. (i) As M is τ -sequentially compact and {xn } is a sequence in M, so {xn } has a convergent subsequence {xm } such that xm → y ∈ M. As I and S, T are continuous and xm = Ixm = Am xm = km Axm + (1 − km )p, xm = J xm = Sm xm = km Sxm + (1 − km )p, so it follows that y = T y = Sy = Iy = J y. (ii) The sequence {xn } has a subsequence {xm } converges to u ∈ M. Since I is weakly continuous and so as in (i), we have Iu = u. Now, xm = Ixm = Am xm = km Axm + (1 − km )p implies that Ixm − Axm = (1 − km )[p − Axm ] → 0 as m → ∞. The demiclosedness of I − A at 0 implies that (I − A)u = 0. Hence Iu = u = Au. Similarly, we can show Su = u = J u,when J − S is demiclosed at 0. This completes the proof.  An immediate consequence of the Theorem 3.3 is as follows: Corollary 3.4. Let M be a nonempty τ -bounded, τ -sequentially complete and p−starshaped subset of a Hausdorff locally convex space (E, τ ). Let A, S, I and J be self mappings of M with A(M) ⊆ J (M) and S(M) ⊆ I(M). Suppose (A, I) and (S, J ) are compatible pairs, A and S are continuous, I and J are nonexpansive, and affine, I(M) = M = J (M), p ∈ F(I) ∩ F(J ). If A, S, I and J satisfy the following: (3.10)

pα (Ax − Sy) < N(x, y)

where N(x, y) = max{pα (Ix − J y), pα (Ix − Ax), pα (J y − Sy), 1 2 pα (Ix

− Sy), 12 pα (J y − Ax)}

for all x, y ∈ M and pα ∈ A∗ (τ ), then A, S, I and J have a common fixed point in M under each of the conditions (i) − (ii) of Theorem 3.3. An immediate consequence of the Theorem 3.3 and Corollary 3.4 is as follows:

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8

M. S. KHAN AND HEMANT KUMAR NASHINE

Corollary 3.5. Let M be a nonempty τ -bounded, τ -sequentially complete and p−starshaped subset of a Hausdorff locally convex space (E, τ ). Let A, S, I and J be self mappings of M with A(M) ⊆ J (M) and S(M) ⊆ I(M). Suppose A, I, S and J are commutative, A and S are continuous, I and J are nonexpansive, and affine, I(M) = M = J (M), p ∈ F(I) ∩ F(J ). If A, S, I and J satisfy (3.4) or (3.10) for all x, y ∈ M and pα ∈ A∗ (τ ), then A, S, I and J have a common fixed point in M under each of the conditions (i) − (ii) of Theorem 3.3. An application of Theorem 3.3, we prove the following more general result in invariant approximation theory: Theorem 3.6. Let A, S, I and J be selfmaps of a Hausdorff locally convex space (E, τ ) and M a subset of E such that A, S(∂M) ⊆ M, where ∂M stands for the boundary of M and x0 ∈ F(A) ∩ F(S) ∩ F(I) ∩ F(J ). Suppose that A and S are continuous, (A, I) and (S, J ) are compatible pairs, I and J are nonexpansive and affine on D = PM (x0 ). Further, suppose A, S, I and J satisfy (3.4) for each x, y ∈ D, pα ∈ A∗ (τ ). If D is nonempty p−starshaped with p ∈ F(I) ∩ F(J ) and I(D) = D = J (D), then A, S, I and J have a common fixed point in D provided one of the following conditions hold: (i) D is τ -sequentially compact; (ii) D is weakly compact in (E, τ ), I and J are weakly continuous and I − A and J − S are demiclosed at 0. Proof. First, we show that A and S are self map on D, i.e., A, S : D → D. Let y ∈ D, then Iy, J y ∈ D, since I(D) = D = J (D). Also, if y ∈ ∂M, then Ay ∈ M, since A(∂M) ⊆ M. Now since Ax0 = Sx0 = x0 = Ix0 = J x0 , so for each pα ∈ A∗ (τ ), we have from (3.4) pα (Ay − x0 ) = pα (Ay − Sx0 ) ≤ N(y, x0 ). Now, Ay ∈ M and Iy ∈ D, this imply that Ay is also closest to x0 , so Ay ∈ D. Similarly Sy ∈ D. Consequently A, S, I and J are selfmaps on D. The conditions of Theorem 3.3 ((i) − (ii)) are satisfied and, hence, there exists a w ∈ D such that Aw = Sw = w = Iw = J w. This completes the proof.  An immediate consequence of the Theorem 3.6 is as follows: Corollary 3.7. Let A, S, I and J be selfmaps of a Hausdorff locally convex space (E, τ ) and M a subset of E such that A, S(∂M) ⊆ M, where ∂M stands for the boundary of M and x0 ∈ F(A) ∩ F(S) ∩ F(I) ∩ F(I). Suppose that A, S are continuous, (A, I) and (S, J ) are compatible pairs, I and J are nonexpansive and affine on D = PM (x0 ). Further, suppose A, S, I and J satisfy (3.4) for each x, y ∈ D, pα ∈ A∗ (τ ). If D is nonempty p−starshaped with p ∈ F(I) ∩ F(J ) and I(D) = D = J (D), then A, S, I and J have a common fixed point in D under each of the conditions (i) − (ii) of Theorem 3.6.

15

ON INVARIANT APPROXIMATION FOR NONCOMMUTATIVE MAPPINGS.....

9

An immediate consequence of the Theorem 3.6 and Corollary 3.7 is as follows: Corollary 3.8. Let A, S, I and J be selfmaps of a Hausdorff locally convex space (E, τ ) and M a subset of E such that A, S(∂M) ⊆ M, where ∂M stands for the boundary of M and x0 ∈ F(A) ∩ F(S) ∩ F(I) ∩ F(I). Suppose that A, S are continuous, A, I, S and J are commutative, I and J are nonexpansive and affine on D = PM (x0 ). Further, suppose A, S, I and J satisfy (3.4) or (3.10) for each x, y ∈ D, pα ∈ A∗ (τ ). If D is nonempty p−starshaped with p ∈ F(I) ∩ F(J ) and I(D) = D = J (D), then A, S, I and J have a common fixed point in D under each of the conditions (i) − (ii) of Theorem 3.6. Remark 3.9. With the remark given by Jungck [4] that every commuting pair of mappings is compatible but the converse is not true in general, and by using four mappings as opposed to three, our results generalize the results of Nashine [10] and consequently other related results (see in [1, 2, 3, 6, 9, 12, 13, 14, 15, 17]). References [1] B. Brosowski, Fixpunkts¨ atze in der Approximationstheorie, Mathematica(Cluj) 11 (1969), 165 - 220. [2] A. Carbone. Some results on invariant approximation. Internat. J. Math. Math. Soc. (3) 17 (1994), 483 - 488. [3] T. L. Hicks and M. D. Humpheries. A note on fixed point theorems. J. Approx. Theory 34 (1982), 221-225. [4] G. Jungck. Compatible mappings and common fixed points. Internat. J. Math. Math. Sci. (4) 9(1986), 771-779. [5] G. Jungck. Common fixed points for commuting and compactible maps on compacta. Proc. Amer. Math. Soc. 103(1988), 977-993. [6] G. Jungck and S. Sessa. Fixed point theorems in best approximation theory. Math. Japonica. (2) 42(1995), 249-252. [7] A. R. Khan and N. Hussain. An extension of a theorem of Sahab, Khan and Sessa. Internat. J. Math. Math. Soc. 27 (11) (2001), 701-706. [8] G. K¨ othe. Topological vector spaces I. Die Grundlehren der mathematischen Wissenschaften, Vol. 159, Springer-Verlag, New York, 1969. [9] G. Meinardus. Invarianze bei Linearen Approximationen. Arch. Rational Mech. Anal. 14(1963), 301 - 303. [10] Hemant Kumar Nashine. Existence of best approximation result in locally convex space. Kungpook Math. J. 46(2006), 389-397. [11] N. Hussain and A. R. Khan. Common fixed point results in best approximation theory. Applied Math. Letters. 16 (2003), 575-580. [12] S. A. Sahab, M. S. Khan and S. Sessa. A result in best approximation theory. J. Approx. Theory 55 (1988), 349-351. [13] S. P. Singh. An application of a fixed point theorem to approximation theory. J. Approx. Theory 25(1979), 89 - 90. [14] S. P. Singh. Application of fixed point theorems to approximation theory in: V. Lakshmikantam (Ed.), Applied nonlinear Analysis, Academic Press, New York, 1979. [15] S. P. Singh. Some results on best approximation in locally convex spaces. J. Approx. Theory 28 (1980), 329 - 332. [16] S. P. Singh, B. Watson and P. Srivastava. Fixed point theory and best approximation: The KKM-Map Principle Vol. 424, Kluwer Academic Publishers, 1997. [17] P. V. Subrahmanyam. An application of a fixed point theorem to best approximations. J. Approx. Theory 20(1977), 165- 172. [18] E. Tarafdar. Some fixed point theorems on locally convex linear topological spaces. Bull. Austral. Math. Soc. 13(1975), 241-254.

16

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.1,17-23,2008, COPYRIGHT 2008 EUDOXUS PRESS, 17 LLC

A new application of almost increasing sequences H¨ useyin Bor Department of Mathematics, Erciyes University, 38039 Kayseri, TURKEY E-mail:[email protected], URL:http://fef.erciyes.edu.tr/math/hbor.htm

Abstract ¯ , pn | summability factors of infinite series has In the present paper, a theorem on | N k been proved under weaker conditions. Also we have obtained a new result concerning the | C, 1 |k summability factors.

1 Let

Introduction P

an be a given infinite series with partial sums (sn ). We denote by uαn and tαn the

n-th Ces`aro means of order α, with α > −1, of the sequence (sn ) and (nan ), respectively, i.e., uαn

n 1 X = α Aα−1 sv , An v=0 n−v

(1)

n 1 X Aα−1 vav , Aαn v=1 n−v

(2)

tαn = where Aαn = O(nα ), The series

P

α > −1,

Aα0 = 1 and Aα−n = 0 f or

n > 0.

(3)

an is said to be summable | C, α |k , k ≥ 1, if (see [5]) ∞ X

k−1

n

|

uαn

−

uαn−1 |k =

n=1

∞ X | tαn |k v=1

n

< ∞,

2000 AMS Subject Classification: 40D15, 40F05, 40G99. Key Words: Absolute summability, almost increasing sequence. 1

(4)

18

BOR

where tαn = n(uαn − uαn−1 ) (see [7]). Let (pn ) be a sequence of positive numbers such that Pn =

n X

pv → ∞ as

n → ∞,

(P−i = p−i = 0, i ≥ 1).

(5)

v=0

The sequence-to-sequence transformation n 1 X σn = pv sv Pn v=0

(6)

¯ , pn ) mean of the sequence defines the sequence (σn ) of the Riesz mean or simply the (N (sn ), generated by the sequence of coefficients (pn ) (see [6]). The series

P

an is said to be

¯ , pn | , k ≥ 1, if (see [2], [3]) summable | N k ∞ X

(Pn /pn )k−1 | ∆σn−1 |k < ∞,

(7)

n=1

where ∆σn−1 = −

n X pn Pv−1 av , Pn Pn−1 v=1

n ≥ 1.

(8)

¯ , pn | summability is the same as | C, 1 | In the special case pn = 1 for all values of n, | N k k summability. 2. Known Results. Mishra and Srivastava [9] have proved the following theorem for ¯ , pn | summability. |N Theorem A. Let (Xn ) be a positive non-decreasing sequence and let there be sequences (βn ) and (λn ) such that | ∆λn |≤ βn , βn → 0 as ∞ X

(9)

n → ∞,

(10)

n | ∆βn | Xn < ∞,

(11)

n=1

| λn | Xn = O(1). If

n X | sv | v=1

v

= O(Xn ) as 2

(12)

n→∞

(13)

ALMOST INCREASING SEQUENCES

19

and (pn ) is a sequence such that

then the series

P∞

Pn λn n=1 an npn

Pn = O(npn ),

(14)

Pn ∆pn = O(pn pn+1 ),

(15)

¯ , pn |. is summable | N

¯ , pn | summability in the following form. Later on Bor [4] has proved Theorem A for | N k Theorem B. Let (Xn ) be a positive non-decreasing sequence and the sequences (βn ) and (λn ) are such that conditions (9)-(15) of Theorem A are satisfied with the condition (13) replaced by:

n X | sv |k v=1

Then the series

P∞

Pn λn n=1 an npn

= O(Xn ) as

v

n → ∞.

(16)

¯ , pn | , k ≥ 1. is summable | N k

It may be noted that if we take k = 1 in Theorem B, then we get Theorem A. 3. Main result. The aim of this paper is to prove Theorem B under weaker conditions. For this we need the concept of almost increasing sequence. A positive sequence (bn ) is said to be almost increasing if there exists a positive increasing sequence (cn ) and two positive constants A and B such that Acn ≤ bn ≤ Bcn (see [1]). Obviously every increasing sequence is almost increasing. However, the converse need not be true as can be seen by n

taking the example, say bn = ne(−1) . Now, we shall prove the following theorem. Theorem. Let (Xn ) be an almost increasing sequence. If the conditions (9)-(12) and (14)-(16) are satisfied, then the series

P∞

P n λn n=1 an npn

¯ , pn | , k ≥ 1. is summable | N k

Remark. It should be noted that, from the hypotheses of the Theorem, (λn ) is bounded and ∆λn = O(1/n) (see [4]). We require the following lemma for the proof of the theorem. Lemma ([8]). If (Xn ) be an almost increasing sequence, then under the conditions (10)-(11) we have that nXn βn = O(1), ∞ X

βn Xn < ∞.

n=1

3

(17) (18)

20

BOR

¯ , pn ) mean of the series 4. Proof of the Theorem. Let (Tn ) be the sequence of (N P∞

n=1

an Pn λn npn .

Then, by definition, we have Tn =

n v n X 1 X ar Pr λr 1 X av Pv λv pv = (Pn − Pv−1 ) . Pn v=1 r=1 rpr Pn v=1 vpv

Then

n X pn Pv−1 Pv av λv , n ≥ 1. Pn Pn−1 v=1 vpv

Tn − Tn−1 =

(19)

(20)

Using Abel’s transformation, we get Tn − Tn−1 = = +

µ ¶ n X pn Pv−1 Pv λv λn sn sv ∆ + Pn Pn−1 v=1 vpv n X Pv+1 Pv ∆λv pn n−1 sn λn + sv n Pn Pn−1 v=1 (v + 1)pv+1

µ ¶ X X pn n−1 pn n−1 1 Pv Pv sv λv ∆ − sv Pv λv Pn Pn−1 v 1 vpv Pn Pn−1 v=1 v =

= Tn,1 + Tn,2 + Tn,3 + Tn,4 ,

say.

To prove the theorem, by Minkowski’s inequality, it is sufficient to show that ¶ ∞ µ X Pn k−1 n=1

pn

| Tn,r |k < ∞,

f or

r = 1, 2, 3, 4.

(21)

Firstly by using Abel’s transformation, we have that ¶ m µ X Pn k−1 n=1

pn

| Tn,1 |k =

¶ m µ X Pn k−1 n=1

= O(1) = O(1)

npn m X n=1 m−1 X

| λn |

| λn |k−1 | λn | | sn |k n

∆ | λn |

n=1

+ O(1) | λm |

m X | n=1

= O(1) = O(1)

m−1 X n=1 m−1 X

| sn |k n

n X | sv |k

v=1 sn |k

v

n

| ∆λn | Xn + O(1) | λm | Xm βn Xn + O(1) | λm | Xm = O(1) as m → ∞,

n=1

4

ALMOST INCREASING SEQUENCES

21

by virtue of the hypotheses of the Theorem and Lemma. Now, using the fact that Pv+1 = O((v + 1)pv+1 ) by (14), we have m+1 Xµ n=2

Pn pn

¶k−1

| Tn,2 |k = O(1)

m+1 X n=2

= O(1)

m+1 X n=2

n−1 X pn | Pv sv ∆λv |k k Pn Pn−1 v=1

pn k Pn Pn−1

(n−1 X Pv v=1

pv

)k

| sv | pv | ∆λv |

Now applying H¨older’s inequality, we have that m+1 Xµ n=2

Pn pn

¶k−1

k

| Tn,2 |

= O(1) Ã

×

m+1 X n=2

X µ Pv ¶k pn n−1 | sv |k pv | ∆λv |k Pn Pn−1 v=1 pv

1

n−1 X

Pn−1

v=1

= O(1) = O(1) = O(1) = O(1) = O(1) = O(1) = O(1)

!k−1

pv

¶ m µ X Pv k

pv v=1 m µ X Pv v=1 m X

| sv |k pv | ∆λv |k

| ∆λv | pv

¶k−1 µ

k

| sv | | ∆λv |

v=1 m X

vβv

v=1 m−1 X v=1 m−1 X v=1 m−1 X

m+1 X

pn P P n=v+1 n n−1

| sv |k | ∆λv |

Pv vpv

¶k−1

| sv |k v

∆(vβv )

v X | sr |k r=1

r

+ O(1)mβm

m X | sv |k v=1

v

| ∆(vβv ) | Xv + O(1)mβm Xm v | ∆βv | Xv + O(1)

v=1

m−1 X

| βv | Xv + O(1)mβm Xm = O(1)

v=1

as m → ∞, in view of the hypotheses of the Theorem and Lemma. Pv Again, since ∆( vp ) = O( v1 ), by (14) and (15) (see [9]), as in Tn,1 we have that v m+1 Xµ n=2

Pn pn

¶k−1

k

| Tn,3 |

= O(1)

m+1 X n=2

= O(1)

m+1 X n=2

pn k Pn Pn−1 pn k Pn Pn−1 5

(n−1 X

1 Pv | sv || λv | v v=1

(n−1 µ ¶ X Pv v=1

pv

)k

1 pv | sv || λv | v

)k

22

BOR

= O(1)

n=2

(

×

m+1 X

X µ Pv ¶k pn n−1 pv | sv |k | λv |k Pn Pn−1 v=1 vpv

1

n−1 X

Pn−1

v=1

= O(1) = O(1) = O(1) = O(1) = O(1)

)k−1

pv

¶ m µ X Pv k v=1 m µ X v=1 m µ X v=1 m X

vpv Pv vpv Pv vpv

| λv |

v=1 m−1 X

¶k

m+1 X

| sv |k pv | λv |k

pn P P n=v+1 n n−1

pv | sv |k | λv |k

1 v . Pv v

¶k−1

| λv |k−1 | λv |

| sv |k v

| sv |k v

Xv βv + O(1)Xm | λm |= O(1) as m → ∞.

v=1

Finally, using H¨older’s inequality, as in Tn,3 we have m+1 Xµ n=2

Pn pn

¶k−1

k

| Tn,4 |

= = ≤

m+1 X n=2 m+1 X n=2 m+1 X n=2

Ã

×

n−1 X Pv pn | sv λv |k k v Pn Pn−1 v=1 n−1 X pn Pv pv λv |k | sv k vp Pn Pn−1 v=1 v

¶ µ X pn n−1 Pv k k | sv | pv | λv |k Pn Pn−1 v=1 vpv

1

n−1 X

Pn−1

v=1

= O(1) = O(1) = O(1)

!k−1

pv

¶ m µ X Pv k v=1 m X

vpv | λv |

v=1 m−1 X

| sv |k pv | λv |k

1 v . Pv v

| sv |k v

Xv βv + O(1)Xm | λm |= O(1) as m → ∞.

v=1

Therefore we get ¶ m µ X Pn k−1 n=1

pn

| Tn,r |k = O(1) as

m → ∞, 6

f or

r = 1, 2, 3, 4.

ALMOST INCREASING SEQUENCES

23

This completes the proof of the Theorem. Finally if we take pn = 1 for all values of n in this theorem, then we get a new result concerning the | C, 1 |k summability factors.

References [1] S.Aljancic and D.Arandelovic, O-regularly varying functions, Publ. Inst. Math. 22 (1977), 5-22. [2] H.Bor, On two summability methods, Math. Proc. Camb. Philos Soc. 97 (1985), 147-149. [3] H.Bor, A note on two summability methods, Proc. Amer. Math. Soc. 98 (1986), 81-84. ¯ , pn | summability factors of infinite series, Indian J. Pure Appl. [4] H.Bor, A note on | N k Math. 18 (1987), 330-336. [5] T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957), 113-141. [6] G.H.Hardy, Divergent Series, Oxford Univ. Press. Oxford (1949). [7] E. Kogbetliantz, Sur les s´eries absolument sommables par la m´ethode des moyennes arithm´etiques, Bull. Sci. Math. 49 (1925), 234-256. [8] S. M. Mazhar, A note on absolute summability factors, Bull. Inst. Math. Acad. Sinica 25 (1997), 233-242. ¯ , pn | summability factors of infinite [9] K. N. Mishra and R. S. L. Srivastava, On | N series, Indian J. Pure Appl. Math. 15 (1984), 651-656.

7

24

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.1,25-37,2008, COPYRIGHT 2008 EUDOXUS PRESS, 25 LLC

On ∗-Homomorphisms between JC ∗-Algebras Choonkil Park, Won-Gil Park and Hee-Jeong Wee Choonkil Park: Department of Mathematics, Hanyang University, Seoul 133–791, Republic of Korea, [email protected] Won-Gil Park: National Institute for Mathematical Sciences, Daejeon 305–340, Republic of Korea, [email protected] Hee-Jeong Wee: Department of Mathematics, Chungnam National University, Daejeon 305– 764, Republic of Korea, [email protected] Abstract. It is shown that every almost unital almost linear mapping f : A → B of JC ∗ -algebra A to a JC ∗ -algebra B is a homomorphism when f (2n u ◦ y) = f (2n u) ◦ f (y) holds for all unitaries u ∈ A, all y ∈ A, and all n = 0, 1, 2, · · ·, and that every almost unital almost linear continuous mapping f : A → B of a JC ∗ -algebra A of real rank zero to a JC ∗ -algebra B is a homomorphism when f (2n u◦y) = f (2n u)◦f (y) holds for all u ∈ {v ∈ A | v = v ∗ , kvk = 1, v is invertible}, all y ∈ A, and all n = 0, 1, 2, · · ·. Furthermore, we are going to prove the generalized Hyers–Ulam–Rassias stability of ∗-homomorphisms between JC ∗ -algebras, and C-linear ∗-derivations on JC ∗ -algebras. 2000 Mathematics Subject Classification. Primary 47E10, 39B52, 17Cxx, 46L05. Key words and Phrases. Hyers–Ulam–Rassias stability, homomorphism in JC ∗ -algebra, real rank 0, linear derivation

1. Introduction Our knowledge concerning the continuity properties of epimorphisms on Banach algebras, Jordan–Banach algebras, and, more generally, nonassociative complete normed algebras, is now fairly complete and satisfactory (see [9] and [10]). A basic continuity problem consists in determining algebraic conditions on a Banach algebra A which ensure ———————– This work was supported by Korea Research Foundation Grant KRF-2005-041-C00027.

1

26

PARK ET AL

that derivations on A are continuous. In 1996, Villena [10] proved that derivations on semisimple Jordan–Banach algebras are continuous. Let E1 and E2 be Banach spaces with norms || · || and k · k, respectively. Consider f : E1 → E2 to be a mapping such that f (tx) is continuous in t ∈ R for each fixed x ∈ E1 . Assume that there exist constants θ ≥ 0 and p ∈ [0, 1) such that kf (x + y) − f (x) − f (y)k ≤ θ(||x||p + ||y||p ) for all x, y ∈ E1 . Rassias [8] showed that there exists a unique R-linear mapping T : E1 → E2 such that

2θ ||x||p p 2−2 for all x ∈ E1 . G˘avruta [2] generalized the Rassias’ result. kf (x) − T (x)k ≤

Jun, Kim and Shin [4] proved the following: Let X and Y be Banach spaces. Denote by ϕ : X × X → [0, ∞) a function such that ε(x) :=

∞ X

2−j (ϕ(2j−1 x, 0) + ϕ(0, 2j−1 x) + ϕ(2j−1 x, 2j−1 x)) < ∞

j=1

for all x ∈ X. Suppose that f, g, h : X → Y are mappings satisfying k2f (

x+y ) − g(x) − h(y)k ≤ ϕ(x, y) 2

for all x, y ∈ X. Then there exists a unique additive mapping T : X → Y such that x k2f ( ) − T (x)k ≤ kg(0)k + kh(0)k + ε(x), 2 kg(x) − T (x)k ≤ kg(0)k + 2kh(0)k + ϕ(x, 0) + ε(x), kh(x) − T (x)k ≤ 2kg(0)k + kh(0)k + ϕ(0, x) + ε(x) for all x ∈ X. B.E. Johnson [3], Theorem 7.2 also investigated almost algebra ∗-homomorphisms between Banach ∗-algebras : Suppose that U and B are Banach ∗-algebras which satisfy the conditions of [3], Theorem 3.1. Then for each positive ² and K there is a positive δ such that if T ∈ L(U, B) with kT k < K, kT ∨ k < δ and kT (x∗ )∗ − T (x)k ≤ δkxk (x ∈ U ) then there is a ∗-homomorphism T 0 : U → B with kT − T 0 k < ². Here L(U, B) is the space of bounded linear maps from U into B, and T ∨ (x, y) = T (xy) − T (x)T (y) (x, y ∈ U). See [3] for details.

JC*-ALGEBRAS

27

The original motivation to introduce the class of nonassociative algebras known as Jordan algebras came from quantum mechanics (see [9]). Let H be a complex Hilbert space, regarded as the “state space” of a quantum mechanical system. Let L(H) be the real vector space of all bounded self-adjoint linear operators on H, interpreted as the (bounded) observables of the system. In 1932, Jordan observed that L(H) is a (nonassociative) algebra via the anticommutator product x ◦ y :=

xy+yx . 2

A commutative algebra

X with product x ◦ y is called a Jordan algebra if x2 ◦ (x ◦ y) = x ◦ (x2 ◦ y) holds. A complex Jordan algebra C with product x ◦ y and involution x 7→ x∗ is called a JB ∗ -algebra if C carries a Banach space norm k · k satisfying kx ◦ yk ≤ kxk · kyk and k{xx∗ x}|| = kxk3 . Here {xy ∗ z} := x ◦ (y ∗ ◦ z) − y ∗ ◦ (z ◦ x) + z ◦ (x ◦ y ∗ ) denotes the Jordan triple product of x, y, z ∈ C. A unital Jordan C ∗ -subalgebra of a C ∗ -algebra, endowed with the anticommutator product, is called a JC ∗ -algebra. Throughout this paper, let A be a JC ∗ -algebra with norm || · || and unit e, and B a JC ∗ -algebra with norm k · k and unit e0 . Let U(A) = { u ∈ A | u∗ u = uu∗ = e}, Asa = {x ∈ A | x = x∗ }, and I1 (Asa ) = {v ∈ Asa | kvk = 1, v is invertible}. In this paper, we prove that every almost unital almost linear mapping h : A → B is a homomorphism when h(3n u ◦ y) = h(3n u) ◦ h(y) holds for all u ∈ U (A), all y ∈ A, and all n = 0, 1, 2, · · ·, and that for a JC ∗ -algebra A of real rank zero (see [1]), every almost unital almost linear continuous mapping h : A → B is a homomorphism when h(3n u ◦ y) = h(3n u) ◦ h(y) holds for all u ∈ I1 (Asa ), all y ∈ A, and all n = 0, 1, 2, · · ·. Furthermore, we are going to prove the generalized Hyers–Ulam–Rassias stability of ∗-homomorphisms between JC ∗ -algebras, and C-linear ∗-derivations on JC ∗ -algebras.

2. ∗-homomorphisms between JC ∗-algebras We are going to investigate ∗-homomorphisms between JC ∗ -algebras. Theorem 2.1. Let f, g, h : A → B be mappings satisfying f (0) = 0, g(0) = 0 and h(0) = 0, and let f (2n u ◦ y) = f (2n u) ◦ f (y), g(2n u ◦ y) = g(2n u) ◦ g(y) and h(2n u ◦ y) = h(2n u) ◦ h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, · · ·, for which there exists a function ϕ : A \ {0} × A \ {0} → [0, ∞) such that e ϕ(x, y) :=

∞ X j=0

2−j ϕ(2j−1 x, 2j−1 y) < ∞,

(1)

28

PARK ET AL

k2f (

µx + µy ) − µg(x) − µh(y)k ≤ ϕ(x, y), 2 kf (2n u∗ ) − f (2n u)∗ k ≤ ϕ(2n u, 2n u)

(2) (3)

for all µ ∈ T1 := {λ ∈ C | |λ| = 1}, all u ∈ U(A), all x, y ∈ A, and all n = 0, 1, 2, · · ·. Assume that

f (2n e) = e0 . 2n Then the mappings f, g, h : A → B are ∗-homomorphisms. lim n→∞

(4)

Proof. Put µ = 1 ∈ T1 . It follows from Corollary 2.5 of [4] that there exists a unique additive mapping H : A → B such that x k2f ( ) − H(x)k ≤ ε(x), 2 kg(x) − H(x)k ≤ ϕ(x, 0) + ε(x), kh(x) − H(x)k ≤ ϕ(0, x) + ε(x) for all x ∈ A \ {0}, where ε(x) :=

∞ X

2−j (ϕ(2j−1 x, 0) + ϕ(0, 2j−1 x) + ϕ(2j−1 x, 2j−1 x)) < ∞

j=1

for all x ∈ A \ {0}. The additive mapping H : A → B is given by H(x) = n→∞ lim

1 f (2n x) n 2

for all x ∈ A, and lim 2−n f (2n x) = lim 2−n g(2n x) = lim 2−n h(2n x)

n→∞

n→∞

n→∞

for all x ∈ A. Let fe(x) = 2f ( x2 ) for all x ∈ A, then lim

n→∞

1 e n 1 f (2 x) = lim n f (2n x) n n→∞ 2 2

for all x ∈ A. By the assumption, kf (2n µx) − µf (2n x)k = kf (2n µx) − 12 µg(2n x) − 12 µh(2n x) + 12 µg(2n x) + 12 µh(2n x) − µf (2n x)k ≤ 12 ϕ(2n x, 2n x) + 12 |µ|ϕ(2n x, 2n x) = ϕ(2n x, 2n x)

(5)

JC*-ALGEBRAS

29

for all µ ∈ T1 and all x ∈ A \ {0}. Thus 2−n kf (2n µx) − µf (2n x)k → 0 as n → ∞ for all µ ∈ T1 and all x ∈ A \ {0}. Hence H(µx) = n→∞ lim

f (2n µx) µf (2n x) = lim = µH(x) n→∞ 2n 2n

(6)

for all µ ∈ T1 and all x ∈ A \ {0}. λ Now let λ ∈ C (λ 6= 0) and M an integer greater than 2|λ|. Then | M |
0, does there exist a δ(²) > 0 such that if a mapping h : G1 → G2 satisfies the inequality d(h(x ∗ y), h(x) ¦ h(y)) < δ for all x, y ∈ G1 , then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) < ² for all x ∈ G1 ? ———————– ∗

The corresponding author was supported by the research fund of Hanyang University (HY-2006-N).

1

40

LEE-PARK

If the answer is affirmative, we would say that the equation of homomorphism H(x ∗ y) = H(x) ¦ H(y) is stable. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation? D.H. Hyers [13] gave a first affirmative answer to the question of Ulam for Banach spaces. Let X and Y be Banach spaces. Assume that f : X → Y satisfies kf (x + y) − f (x) − f (y)k ≤ ε for all x, y ∈ X and some ε ≥ 0. Then there exists a unique additive mapping T : X → Y such that kf (x) − T (x)k ≤ ε for all x ∈ X. Let X and Y be Banach spaces with norms || · || and k · k, respectively. Consider f : X → Y to be a mapping such that f (tx) is continuous in t ∈ R for each fixed x ∈ X. Th.M. Rassias [28] introduced the following inequality: Assume that there exist constants θ ≥ 0 and p ∈ [0, 1) such that kf (x + y) − f (x) − f (y)k ≤ θ(||x||p + ||y||p ) for all x, y ∈ X. Th.M. Rassias [28] showed that there exists a unique R-linear mapping T : X → Y such that kf (x) − T (x)k ≤

2θ ||x||p 2 − 2p

for all x ∈ X. The above inequality has provided a lot of influence in the development of what is known as Hyers–Ulam–Rassias stability of functional equations. Beginning around the year 1980 the topic of approximate homomorphisms, or the stability of the equation of homomorphism, was studied by a number of mathematicians. G˘avruta [11] generalized the Rassias’ result. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2], [5]–[6], [10], [11], [14]–[18], [20]–[26], [29]–[33], [36], [37]). We recall some basic facts concerning quasi-Banach spaces and some preliminary results.

ON QUASI-BANACH ALGEBRAS

41

Definition 1.1 ([4, 35]) Let X be a real linear space. A quasi-norm is a real-valued function on X satisfying the following: (1) kxk ≥ 0 for all x ∈ X and kxk = 0 if and only if x = 0. (2) kλxk = |λ| · kxk for all λ ∈ R and all x ∈ X. (3) There is a constant K ≥ 1 such that kx + yk ≤ K(kxk + kyk) for all x, y ∈ X. The pair (X, k · k) is called a quasi-normed space if k · k is a quasi-norm on X. The smallest possible K is called the modulus of concavity of k · k. A quasi-Banach space is a complete quasi-normed space. A quasi-norm k · k is called a p-norm (0 < p ≤ 1) if kx + ykp ≤ kxkp + kykp for all x, y ∈ X. In this case, a quasi-Banach space is called a p-Banach space. Given a p-norm, the formula d(x, y) := kx − ykp gives us a translation invariant metric on X. By the Aoki–Rolewicz theorem [35] (see also [4]), each quasi-norm is equivalent to some p-norm. Since it is much easier to work with p-norms than quasi-norms, henceforth we restrict our attention mainly to p-norms. Definition 1.2 ([1]) Let (A, k · k) be a quasi-normed space. The quasi-normed space (A, k · k) is called a quasi-normed algebra if A is an algebra and there is a constant C > 0 such that kxyk ≤ Ckxk · kyk for all x, y ∈ A. A quasi-Banach algebra is a complete quasi-normed algebra. If the quasi-norm k · k is a p-norm then the quasi-Banach algebra is called a p-Banach algebra. Definition 1.3 Let A and B be quasi-Banach algebras with norms k · kA and k · kB . An algebra homomorphism H : A → B is called an isometric homomorphism if the algebra homomorphism H : A → B satisfies kH(x) − H(y)kB = kx − ykA for all x, y ∈ A. If, in addition, the algebra homomorphism H : A → B is bijective, then the algebra homomorphism H : A → B is called an isometric isomorphism. The stability of isometries in normed spaces and Banach algebras have been investigated in several papers (see [3, 8, 9, 12, 19]).

42

LEE-PARK

The paper is organized as follows: In Section 2, we prove the Hyers–Ulam–Rassias stability of isometric homomorphisms in quasi-Banach algebras, associated to the Cauchy functional equation and the Jensen functional equation. In Section 3, we investigate isometric isomorphisms between quasi-Banach algebras.

2

Stability of isometric homomorphisms in quasi-Banach algebras

Throughout this section, assume that A is a quasi-normed algebra with quasi-norm k · kA and that B is a p-Banach algebra with p-norm k · kB . Let K be the modulus of concavity of k · kB . We prove the Hyers–Ulam–Rassias stability of isometric homomorphisms in quasiBanach algebras, associated to the Cauchy functional equation. Theorem 2.1 Let r > 2 and θ be positive real numbers, and let f : A → B be a mapping such that kf (x + y) − f (x) − f (y)kB ≤ θ(kxkrA + kykrA ),

(1)

kf (xy) − f (x)f (y)kB ≤ θ(kxkrA + kykrA ),

(2)

| kf (x)kB − kxkA | ≤ θkxkrA

(3)

for all x, y ∈ A. If f (tx) is continuous in t ∈ R for each fixed x ∈ A, then there exists a unique isometric homomorphism H : A → B such that kf (x) − H(x)kB ≤

2θ (2pr

−

1

2p ) p

kxkrA

(4)

for all x ∈ A. Proof. Letting y = x in (1), we get kf (2x) − 2f (x)kB ≤ 2θkxkrA for all x ∈ A. So

x 2θ kf (x) − 2f ( )kB ≤ r kxkrA 2 2

(5)

ON QUASI-BANACH ALGEBRAS

43

for all x ∈ A. Since B is a p-Banach algebra, m−1 X X 2pj x x x x 2p θp m−1 k2l f ( l ) − 2m f ( m )kpB ≤ k2j f ( j ) − 2j+1 f ( j+1 )kpB ≤ pr kxkpr A prj 2 2 2 2 2 2 j=l j=l

(6)

for all nonnegative integers m and l with m > l and all x ∈ A. It follows from (6) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ A. Since B is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping H : A → B by H(x) := n→∞ lim 2n f (

x ) 2n

for all x ∈ A. It follows from (1) that kH(x + y) − H(x) − H(y)kB = ≤

lim 2n kf (

n→∞

x+y x y ) − f ( n ) − f ( n )kB n 2 2 2

2n θ (kxkrA + kykrA ) = 0 n→∞ 2nr lim

for all x, y ∈ A. So H(x + y) = H(x) + H(y) for all x, y ∈ A. Moreover, letting l = 0 and passing the limit m → ∞ in (6), we get (4). By the same reasoning as in the proof of Theorem of [28], the mapping H : A → B is R-linear.

It follows from (2) that kH(xy) − H(x)H(y)kB =

lim 4n kf ( n→∞

xy x y ) − f ( )f ( )kB 2n · 2n 2n 2n

4n θ ≤ n→∞ lim nr (kxkrA + kykrA ) = 0 2 for all x, y ∈ A. So H(xy) = H(x)H(y) for all x, y ∈ A.

Now, let T : A → B be another Cauchy additive mapping satisfying (4). Then we have x x ) − T ( )kB 2n 2n x x x x ≤ 2n K(kH( n ) − f ( n )kB + kT ( n ) − f ( n )kB ) 2 2 2 2 2n+2 Kθ ≤ kxkrA , 1 pr p nr p (2 − 2 ) 2

kH(x) − T (x)kB = 2n kH(

44

LEE-PARK

which tends to zero as n → ∞ for all x ∈ A. So we can conclude that H(x) = T (x) for all x ∈ A. This proves the uniqueness of H. It follows from (3) that x 2n θ )k − kxk | = kxkrA , B A 2n 2nr which tends to zero as n → ∞ for all x ∈ A. So x kH(x)kB = n→∞ lim k2n f ( n )kB = kxkA 2 for all x ∈ A. Hence | k2n f (

kH(x) − H(y)kB = kH(x − y)kB = kx − ykA for all x, y ∈ A. So the mapping H : A → B is an isometry. Thus the mapping H : A → B is a unique isometric homomorphism satisfying (4). 2 Theorem 2.2 Let r < 1 and θ be positive real numbers, and let f : A → B be a mapping satisfying (1), (2) and (3). If f (tx) is continuous in t ∈ R for each fixed x ∈ A, then there exists a unique isometric homomorphism H : A → B such that kf (x) − H(x)kB ≤

2θ (2p

−

1

2pr ) p

kxkrA

(7)

for all x ∈ A. Proof. It follows from (5) that 1 kf (x) − f (2x)kB ≤ θkxkrA 2 for all x ∈ A. Since B is a p-Banach algebra, m−1 m−1 X 1 X 2prj 1 1 1 k l f (2l x) − m f (2m x)kpB ≤ k j f (2j x) − j+1 f (2j+1 x)kpB ≤ θp kxkpr A pj 2 2 2 2 2 j=l j=l

(8)

for all nonnegative integers m and l with m > l and all x ∈ A. It follows from (8) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ A. Since B is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping H : A → B by H(x) := lim

n→∞

1 f (2n x) 2n

for all x ∈ A. The rest of the proof is similar to the proof of Theorem 2.1. 2 We prove the Hyers–Ulam–Rassias stability of isometric homomorphisms in quasiBanach algebras, associated to the Jensen functional equation.

ON QUASI-BANACH ALGEBRAS

45

Theorem 2.3 Let r < 1 and θ be positive real numbers, and let f : A → B be a mapping with f (0) = 0 satisfying (2) and (3) such that k2f (

x+y ) − f (x) − f (y)kB ≤ θ(kxkrA + kykrA ) 2

(9)

for all x, y ∈ A. If f (tx) is continuous in t ∈ R for each fixed x ∈ A, then there exists a unique isometric homomorphism H : A → B such that kf (x) − H(x)kB ≤

K(3 + 3r )θ (3p

−

1

3pr ) p

kxkrA

(10)

for all x ∈ A. Proof. Letting y = −x in (9), we get k − f (x) − f (−x)kB ≤ 2θkxkrA for all x ∈ A. Letting y = 3x and replacing x by −x in (9), we get k2f (x) − f (−x) − f (3x)kB ≤ (3r + 1)θkxkrA for all x ∈ A. Thus k3f (x) − f (3x)kB ≤ K(3r + 3)θkxkrA

(11)

for all x ∈ A. So 1 K(3r + 3)θ kf (x) − f (3x)kB ≤ kxkrA 3 3 for all x ∈ A. Since B is a p-Banach algebra, m−1 X 1 1 1 1 k l f (3l x) − m f (3m x)kpB ≤ k j f (3j x) − j+1 f (3j+1 x)kpB 3 3 3 j=l 3

X 3prj K p (3r + 3)p θp m−1 ≤ kxkpr A p pj 3 j=l 3

(12)

for all nonnegative integers m and l with m > l and all x ∈ A. It follows from (12) that the sequence { 31n f (3n x)} is a Cauchy sequence for all x ∈ A. Since B is complete, the sequence { 31n f (3n x)} converges. So one can define the mapping H : A → B by H(x) := n→∞ lim

1 f (3n x) n 3

46

LEE-PARK

for all x ∈ A. By (9), k2H(

x+y ) − H(x) − H(y)kB = 2

1 x+y k2f (3n · ) − f (3n x) − f (3n y)kB n 3 2 3rn r ≤ lim n θ(kxkA + kykrA ) = 0 n→∞ 3 lim

n→∞

for all x, y ∈ A. So

x+y ) = H(x) + H(y) 2 for all x, y ∈ A. Moreover, letting l = 0 and passing the limit m → ∞ in (12), we get 2H(

(10). It follows from (2) that 1 kf (9n xy) − f (3n x)f (3n y)kB 9n 3nr θ ≤ lim n (kxkrA + kykrA ) = 0 n→∞ 9

kH(xy) − H(x)H(y)kB =

lim n→∞

for all x, y ∈ A. So H(xy) = H(x)H(y) for all x, y ∈ A. Now, let T : A → B be another Jensen additive mapping satisfying (10). Then we have 1 kH(3n x) − T (3n x)kpB pn 3 1 ≤ pn (kH(3n x) − f (3n x)kpB + kT (3n x) − f (3n x)kpB ) 3 3prn K p (3 + 3r )p θp ≤ 2 · pn · kxkpr A, p pr 3 3 −3 which tends to zero as n → ∞ for all x ∈ A. So we can conclude that H(x) = T (x) for kH(x) − T (x)kpB =

all x ∈ A. This proves the uniqueness of H. The rest of the proof is similar to the proof of Theorem 2.1. 2 Theorem 2.4 Let r > 2 and θ be positive real numbers, and let f : A → B be a mapping with f (0) = 0 satisfying (2), (3) and (9). If f (tx) is continuous in t ∈ R for each fixed x ∈ A, then there exists a unique isometric homomorphism H : A → B such that kf (x) − H(x)kB ≤ for all x ∈ A.

K(3r + 3)θ (3pr

−

1

3p ) p

kxkrA

(13)

ON QUASI-BANACH ALGEBRAS

47

Proof. It follows from (11) that x K(3r + 3)θ kf (x) − 3f ( )kB ≤ kxkrA 3 3r for all x ∈ A. Since B is a p-Banach algebra, m−1 X x x x x k3l f ( l ) − 3m f ( m )kpB ≤ k3j f ( j ) − 3j+1 f ( j+1 )kpB 3 3 3 3 j=l

≤

X 3pj K p (3r + 3)p θp m−1 kxkpr A pr prj 3 j=l 3

(14)

for all nonnegative integers m and l with m > l and all x ∈ A. It follows from (14) that the sequence {3n f ( 3xn )} is a Cauchy sequence for all x ∈ A. Since B is complete, the sequence {3n f ( 3xn )} converges. So one can define the mapping H : A → B by H(x) := lim 3n f ( n→∞

x ) 3n

for all x ∈ A. The rest of the proof is similar to the proofs of Theorems 2.1 and 2.3. 2

3

Isometric isomorphisms between quasi-Banach algebras

Throughout this section, assume that A is a quasi-Banach algebra with quasi-norm k · kA and unit e and that B is a p-Banach algebra with p-norm k · kB and unit e0 . Let K be the modulus of concavity of k · kB . We investigate isometric isomorphisms between quasi-Banach algebras, associated to the Cauchy functional equation. Theorem 3.1 Let r > 2 and θ be positive real numbers, and let f : A → B be a bijective mapping satisfying (1) and (3) such that f (xy) = f (x)f (y)

(15)

for all x, y ∈ A. If limn→∞ 2n f ( 2en ) = e0 and f (tx) is continuous in t ∈ R for each fixed x ∈ A, then the mapping f : A → B is an isometric isomorphism.

48

LEE-PARK

Proof. Since f (xy) − f (x)f (y) = 0 for all x, y ∈ A, the mapping f : A → B satisfies (2). By Theorem 2.1, there exists an isometric homomorphism H : A → B satisfying (4). The mapping H : A → B is defined by H(x) = lim 2n f ( n→∞

x ) 2n

for all x ∈ A. It follows from (15) that H(x) = H(ex) = lim 2n f ( n→∞

0

ex e e n n ) = lim 2 f ( · x) = lim 2 f ( )f (x) n→∞ n→∞ 2n 2n 2n

= e f (x) = f (x) for all x ∈ A. So the bijective mapping f : A → B is an isometric isomorphism. 2 Theorem 3.2 Let r < 1 and θ be positive real numbers, and let f : A → B be a bijective mapping satisfying (1), (3) and (15). If f (tx) is continuous in t ∈ R for each fixed x ∈ A and limn→∞

1 f (2n e) 2n

= e0 , then the mapping f : A → B is an isometric isomorphism.

Proof. Since f (xy) − f (x)f (y) = 0 for all x, y ∈ A, the mapping f : A → B satisfies (2). By Theorem 2.2, there exists an isometric homomorphism H : A → B satisfying (7). The mapping H : A → B is defined by 1 f (2n x) n→∞ 2n

H(x) = lim for all x ∈ A.

The rest of the proof is similar to the proof of Theorem 3.1. 2 We investigate isometric isomorphisms between quasi-Banach algebras, associated to the Jensen functional equation. Theorem 3.3 Let r < 1 and θ be positive real numbers, and let f : A → B be a bijective mapping with f (0) = 0 satisfying (3), (9) and (15). If f (tx) is continuous in t ∈ R for each fixed x ∈ A and limn→∞

1 f (3n e) 3n

= e0 , then the mapping f : A → B is an isometric

isomorphism. Proof. Since f (xy) − f (x)f (y) = 0 for all x, y ∈ A, the mapping f : A → B satisfies (2). By Theorem 2.3, there exists an isometric homomorphism H : A → B satisfying (10). The mapping H : A → B is defined by H(x) = n→∞ lim

1 f (3n x) n 3

ON QUASI-BANACH ALGEBRAS

49

for all x ∈ A. The rest of the proof is similar to the proof of Theorem 3.1. 2 Theorem 3.4 Let r > 2 and θ be positive real numbers, and let f : A → B be a bijective mapping with f (0) = 0 satisfying (3), (9) and (15). If f (tx) is continuous in t ∈ R for each fixed x ∈ A and limn→∞ 3n f ( 3en ) = e0 , then the mapping f : A → B is an isometric isomorphism. Proof. Since f (xy) − f (x)f (y) = 0 for all x, y ∈ A, the mapping f : A → B satisfies (2). By Theorem 2.4, there exists an isometric homomorphism H : A → B satisfying (13). The mapping H : A → B is defined by H(x) = n→∞ lim 3n f (

x ) 3n

for all x ∈ A. The rest of the proof is similar to the proof of Theorem 3.1. 2

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[10] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [11] P. Gˇavruta, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [12] J. Gevirtz, Stability of isometries on Banach spaces, Proc. Amer. Math. Soc. 89 (1983), 633–636. [13] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [14] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [15] D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125–153. [16] K. Jun, H. Kim and I. Chang, On the Hyers–Ulam stability of an Euler–Lagrange type cubic functional equation, J. Comput. Anal. Appl. 7 (2005), 21–33. [17] K. Jun and Y. Lee, A generalization of the Hyers–Ulam–Rassias stability of Jensen’s equation, J. Math. Anal. Appl. 238 (1999), 305–315. [18] S. Jung,

Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis,

Hadronic Press Inc., Palm Harbor, Florida, 2001. [19] N. Kalton, An elementary example of a Banach space not isomorphic to its complex conjugate, Canad. Math. Bull. 38 (1995), 218–222. [20] C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), 711–720. [21] C. Park, Generalized quadratic mappings in several variables, Nonlinear Anal.–TMA 57 (2004), 713–722. [22] C. Park, Lie ∗-homomorphisms between Lie C ∗ -algebras and Lie ∗-derivations on Lie C ∗ -algebras, J. Math. Anal. Appl. 293 (2004), 419–434. [23] C. Park, Universal Jensen’s equations in Banach modules over a C ∗ -algebra and its unitary group, Acta Math. Sinica 20 (2004), 1047–1056. [24] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [25] C. Park, Cauchy–Rassias stability of a generalized Trif ’s mapping in Banach modules and its applications, Nonlinear Anal.–TMA 62 (2005), 595–613. [26] C. Park, Isomorphisms between unital C ∗ -algebras, J. Math. Anal. Appl. 307 (2005), 753–762. [27] C. Park, Completion of quasi-normed algebras and quasi-normed modules, J. Chungcheong Math. Soc. 19 (2006), 9–18. [28] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.

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[29] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [30] Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378. [31] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [32] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130. [33] Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003. ˇ [34] Th.M. Rassias and P. Semrl, On the Hyers–Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325–338. [35] S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ., Reidel and Dordrecht, 1984. [36] P.K. Sahoo, A generalized cubic functional equation, Acta Math. Sinica 21 (2005), 1159–1166. [37] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [38] J. Tabor, Stability of the Cauchy functional equation in quasi-Banach spaces, Ann. Polon. Math. 83 (2004), 243–255. [39] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960.

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BETTER ERROR ESTIMATION FOR ´ SZASZ-MIRAKJAN-BETA OPERATORS ¨ ¨ ˘ OKTAY DUMAN, MEHMET ALI OZARSLAN, AND HUSEYIN AKTUGLU Abstract. In this paper, we present a modification of a sequence of mixed summation-integral type operators having Sz´ asz and Beta basis functions in summation and integration, the so-called Sz´ asz-MirakjanBeta operators. Then we show that our modified operators have a better error estimation on the interval [0, 2]. Furthermore, we give an r-th order generalization of the modified Sz´ asz-Mirakjan-Beta operators and investigate their approximation properties.

1. Introduction The classical Sz´asz-Mirakjan operators are defined by µ ¶ ∞ X (nx)k k −nx f , Sn (f ; x) := e k! n k=0

where f ∈ C[0, ∞), x ≥ 0 and n ∈ N. Some approximation properties of the Sz´asz-Mirakjan operators and their ¨ modifications were studied by Agrawal and Kasana [1], Duman and Ozarslan [3], Finta [4], Finta, Govil and Gupta [5], Gupta [6], Gupta and Noor [7], Gupta, Noor and Beniwal [8], Gupta and Pant [9], Srivastava and Gupta [11], Totik [12], Zeng and Piriou [13]. Further properties and general approximation results on these operators may be found in the monograph by Altomore and Campiti [2]. Recently Gupta and Noor [7] have proposed a sequence of mixed summationintegral type operators, the so-called Sz´asz-Mirakjan-Beta operators, as follows: (1.1) Z ∞ ∞ X (nx)k tk−1 −nx Un (f ; x) = e f (t) dt + e−nx f (0), k!B(n + 1, k) 0 (1 + t)n+k+1 k=1

where f ∈ C[0, ∞) such that |f (t)| ≤ M (1 + t)γ for some M > 0, γ > 0. Now, for the operators Un given by (1.1), the following lemma follows from [7] immediately. Key words and phrases. Sz´ asz-Mirakjan-Beta operators; positive linear operators; the Korovkin-type approximation theorem; modulus of continuity; Lipschitz class functionals. 2000 Mathematics Subject Classification. 41A25, 41A36. 1

54

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¨ ¨ ˘ OKTAY DUMAN, MEHMET ALI OZARSLAN, AND HUSEYIN AKTUGLU

Lemma A [7]. Let ei (x) = xi , i = 0, 1, 2. Then, for each x ≥ 0 and n > 1, we have (a) Un (e0 ; x) = 1, (b) Un (e1 ; x) = x, ¢ 1 ¡ 2 (c) Un (e2 ; x) = nx + 2x . n−1 Lemma A shows that the operators Un preserve the test functions e0 (x) = 1 and e1 (x) = x. Actually, many well-known approximating operators preserve these test functions, such as Bernstein polynomials, Meyer-K¨ onig and Zeller operators, Sz´asz-Mirakjan operators, Baskakov operators etc. Observe that these operators do not preserve the test function e2 (x) = x2 . However, by modifying the Bernstein polynomials, King [10] presented a non-trivial sequence of positive linear operators which approximate each continuous function on [0, 1] while preserving the functions e0 and e2 . Then it is proved that these modified operators have a better rate of convergence than the classical Bernstein polynomials on the interval [0, 1/3]. Thus a natural question arises: can we construct a sequence of positive linear operators preserving the test functions e0 and e2 so that our modified operators have a better error estimation than Sz´asz-Mirakjan-Beta operators. In the present paper we mainly focus on this problem. 2. Construction of the operators We first consider the Banach lattice Cγ [0, ∞) := {f ∈ C[0, +∞) : |f (x)| ≤ M (1 + x)γ for some M > 0, γ > 0} endowed with the norm kf kγ :=

|f (x)| . γ x∈[0,+∞) (1 + x) sup

Then, the set {e0 , e1 , e2 } is a K+ −subset of Cγ [0, ∞); also the space Cγ [0, ∞) is isomorphic to C[0, 1] (see, for details, [2]). Let {rn (x)} be sequence of real-valued continuous functions defined on [0, ∞) with 0 ≤ rn (x) < ∞. Then we have Z ∞ ∞ X (nrn (x))k tk−1 Un (f ; rn (x)) = e−nrn (x) f (t) dt +e−nrn (x) f (0), k!B(n + 1, k) 0 (1 + t)n+k+1 k=1

where x ∈ [0, ∞), f ∈ Cγ [0, ∞), γ > 0 and n ∈ N. Now, if we replace rn (x) by rn∗ (x) defined as ´ p 1³ (2.1) rn∗ (x) := −1 + 1 + n(n − 1)x2 , x ≥ 0 and n ∈ N, n then we get the following positive linear operators (2.2) Z ∞ ∞ X (nrn∗ (x))k tk−1 ∗ ∗ Un∗ (f ; x) := e−nrn (x) f (t) dt +e−nrn (x) f (0), k!B(n + 1, k) 0 (1 + t)n+k+1 k=1

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´ SZASZ-MIRAKJAN-BETA OPERATORS

3

where f ∈ Cγ [0, ∞), γ > 0 and x ≥ 0. Then, observe that every Un∗ maps CB [0, +∞), the space of all bounded and continuous functions on [0, +∞), into itself. On the other hand, from Lemma A we obtain the following result at once. Lemma 2.1. For each x ≥ 0, we have (a) Un∗ (e0 ; x) = 1, ´ p 1³ (b) Un∗ (e1 ; x) = −1 + 1 + n(n − 1)x2 , n (c) Un∗ (e2 ; x) = x2 . Now, fix b > 0 and consider the lattice homomorphism Tb : C[0, +∞) → C[0, b] defined by Tb (f ) := f |[0,b] for every f ∈ C[0, +∞). In this case, we see that, for each i = 0, 1, 2, (2.3)

lim Tb (Un∗ (ei )) = Tb (ei ) uniformly on [0, b].

n→∞

Thus, with the universal Korovkin-type property with respect to monotone operators (see Theorem 4.1.4 (vi) of [2, p. 199]) we have the following: “Let X be a compact set and H be a cofinal subspace of C(X). If E is a Banach lattice, S : C(X) → E is a lattice homomorphism and if {Ln } is a sequence of positive linear operators from C(X) into E such that limn→∞ Ln (h) = S(h) for all h ∈ H, then limn→∞ Ln (f ) = f provided that f belongs to the Korovkin closure of H”. Hence, by using (2.3) and the above property we obtain the following Korovkin-type approximation result. Theorem 2.2. limn→∞ Un∗ (f ; x) = f (x) uniformly with respect to x ∈ [0, b] provided f ∈ Cγ [0, ∞), γ > 0 and b > 0. 3. Better error estimation In this section we compute the rate of convergence of the operators Un∗ defined by (2.2). Then, we will show that our operators has better error estimation on the interval [0, 2] than that of the Sz´asz-Mirakjan-Beta operators Un given by (1.1). To achieve this we use the modulus of continuity and the elements of Lipschitz class functionals. If we define the function ψx , (x ≥ 0), by ψx (y) = y − x, then by Lemma 2.1 one can get the following result, immediately. Lemma 3.1. For every x ≥ 0, we have ´ p 1³ ∗ 2 (a) Un (ψx ; x) = −x + −1 + 1 + n(n − 1)x , Ã n ! p 1 + n(n − 1)x2 1 2 ∗ (b) Un (ψx ; x) = 2x x + − . n n Let f ∈ CB [0, +∞) and x ≥ 0. Then, the modulus of continuity of f denoted by ω(f, δ), is defined to be ω(f, δ) =

sup |y−x|≤δ; x,y∈[0,+∞)

|f (y) − f (x)| .

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¨ ¨ ˘ OKTAY DUMAN, MEHMET ALI OZARSLAN, AND HUSEYIN AKTUGLU

4

Then we have the following Theorem 3.2. For every f ∈ CB [0, +∞), x ≥ 0 and n > 1, we have where δn,x

|Un∗ (f ; x) − f (x)| ≤ 2ω(f, δn,x ), p := 2x (x − rn∗ (x)) and rn∗ (x) is given by (2.1).

Proof. Now, let f ∈ CB [0, +∞) and x ≥ 0. Using linearity and monotonicity of Un∗ we easily get, for every δ > 0 and n ∈ N, that ½ ¾ 1p ∗ 2 ∗ Un (ψx ; x) . |Un (f ; x) − f (x)| ≤ ω(f, δ) 1 + δ Now applying Lemma 3.1 (b) and choosing δ = δn,x the proof is completed. ¤ Remark. For the Sz´asz-Mirakjan-Beta operators given by (1.1) we may write that, for every f ∈ CB [0, +∞), x ≥ 0 and n > 1, (3.1)

q

|Un (f ; x) − f (x)| ≤ 2ω(f, αn,x ),

where αn,x := x(2+x) n−1 (see [7]). Now we claim that the error estimation in Theorem 3.2 is better than that of (3.1) provided f ∈ CB [0, +∞) and x ∈ [0, 2]. Indeed, for 0 ≤ x ≤ 2, 2 we have x4 ≤ 1. Also since (n − 12 )2 − n(n − 1) = 41 , we can write that · ¸ 1 2 2 x (n − ) − n(n − 1) ≤ 1, 2 or 1 1 + n(n − 1)x2 ≥ (n − )2 x2 2 which gives µ ¶ p 2n − 1 2 1 + n(n − 1)x ≥ x. 2 Then we obtain µ ¶ 1 1p 1 2n − 1 − + 1 + n(n − 1)x2 ≥ − + x. n n n 2n Using the above inequality we have x − rn∗ (x) ≤ or

2+x 2n

x(2 + x) x(2 + x) ≤ n n−1 for x ∈ [0, 2] and n > 1. This guarantees that δn,x ≤ αn,x for x ∈ [0, 2] and n > 1, which corrects our claim. Now we can also compute the rate of convergence of the operators Un∗ by means of the elements of the Lipschitz class LipM (α), (α ∈ (0, 1]). To

(3.2)

2x (x − rn∗ (x)) ≤

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´ SZASZ-MIRAKJAN-BETA OPERATORS

5

get this, we recall that a function f ∈ CB [0, ∞) belongs to LipM (α) if the inequality |f (y) − f (x)| ≤ M |y − x|α (x, y ∈ [0, ∞))

(3.3) holds.

Theorem 3.3. For every f ∈ LipM (α), x ≥ 0 and n > 1, we have α

|Un∗ (f ; x) − f (x)| ≤ M {2x (x − rn∗ (x))} 2 , where rn∗ (x) is given by (2.1). Proof. Since f ∈ LipM (α) and x ≥ 0, using inequality (3.3) and then apply2 ing the H¨older inequality with p = α2 , q = 2−α we get |Un∗ (f ; x) − f (x)| ≤ Un∗ (|f (y) − f (x)| ; x) ≤ M Un∗ (|y − x|α ; x) © ¡ ¢ª α ≤ M Un∗ ψx2 ; x 2 α ≤ M {2x (x − rn∗ (x))} 2 , whence the result.

¤

Notice that as in the proof of Theorem 3.2, since Un (ψx2 ; x) = Sz´ asz-Mirakjan-Beta operators defined by (1.1) satisfy ½ ¾α x(2 + x) 2 (3.4) |Un (f ; x) − f (x)| ≤ M n−1

x(2+x) n−1 ,

the

for every f ∈ LipM (α), x ≥ 0 and n > 1. So, it follows from (3.2) that the above claim also holds for Theorem 3.2, i.e., the rate of convergence of the operators Un∗ by means of the elements of the Lipschitz class functionals is better than the ordinary error estimation given by (3.4) whenever x ∈ [0, 2]. 4. r − th order generalization of the operators Un∗ (r)

Let Cγ [0, ∞), r = 0, 1, 2, ..., denote the space of all functions f ∈ Cγ [0, ∞) such that the r-th derivative f (r) ∈ Cγ [0, ∞) (for some γ > 0) (0) with f (0) (x) := f (x). In the case of r = 0, the space Cγ [0, ∞) coincides with Cγ [0, ∞). Now we consider the following r−th order generalization of the positive linear operators Un∗ defined by (2.2): ∗

∗ (f ; x) = e−nrn (x) Un,r

(4.1) ∗

(nrn∗ (x))k R∞ (i) (t − x)i f (t) dt, i! k=0 i=0 k!B(n + 1, k) 0

+e−nrn (x) (r)

∞ P r P

r (−1)i xi f i (0) P i! i=0

where f ∈ Cγ [0, ∞), γ > 0, r = 0, 1, 2, ..., n ∈ N and rn∗ (x) is given by ∗ = U ∗. (2.1). Observe that Un,0 n

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¨ ¨ ˘ OKTAY DUMAN, MEHMET ALI OZARSLAN, AND HUSEYIN AKTUGLU

6

∗ we may write that Now using the definition of the operators Un,r Z ∞X r (t − x)i ∗ (4.2) Un,r (f ; x) = Wn (x, t)f (i) (t) dt, i! 0 i=0

where Wn (x, t) = e

∗ (x) −nrn

∞ X k=1

(nrn∗ (x))k tk−1 ∗ + e−nrn (x) δ(t) k!B(n + 1, k) (1 + t)n+k+1

and δ(t) is the Dirac delta function. Thus we have the following (r)

Theorem 4.1. For all f ∈ Cγ [0, ∞), γ > 0, such that f (r) ∈ LipM (α), and for every x ≥ 0 we have ¯ ∗ ¯ ¯ ¡ ¢¯ α ¯Un,r (f ; x) − f (x)¯ ≤ M B(α, r) ¯Un∗ |t − x|r+α ; x ¯ , (r − 1)! α + r where r = 1, 2, ... and B(α, r) is the beta function. Proof. By (4.2) and Lemma 2.1 (a) one can write that ( ) Z ∞ r X (t − x)i ∗ (i) (4.3) f (x) − Un,r (f ; x) = Wn (x, t) f (x) − f (t) dt i! 0 i=0

Then we known from Taylor’s formula that (x − t)r (x − t)i = i! (r − 1)! i=0 © ª R1 × (1 − s)r−1 f (r) (t + s(x − t)) − f (r) (t) ds.

f (x) − (4.4)

r P

f (i) (t)

0

Since f (r) ∈ LipM (α), ¯ ¯ ¯ (r) ¯ α (4.5) ¯f (t + s(x − t)) − f (r) (t)¯ ≤ M sα |t − x| . Using (4.5) and the usual definition of the beta integral in (4.4) we conclude that ¯ ¯ r ¯ X α (x − t)i ¯¯ M ¯ (i) (4.6) B(α, r) |t − x|r+α . f (t) ¯f (x) − ¯≤ ¯ i! ¯ (r − 1)! α + r i=0

Thus, the proof is completed by considering (4.3) and (4.6).

¤

∗ given by (4.2) Finally, for the uniform convergence of the operators Un,r we obtain the next result. (r)

Theorem 4.2. For every f ∈ Cγ [0, ∞), γ > 0, r = 1, 2, ..., such that f (r) ∈ LipM (α), we have ∗ lim Un,r (f ; x) = f (x) uniformly with respect to x ∈ [0, b], b > 0.

n→∞

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´ SZASZ-MIRAKJAN-BETA OPERATORS

7

Proof. Let x ∈ [0, b] and define the function g by g(t) = |t − x|r+α . Then, from Theorem 2.2, it is clear that lim Un∗ (g; x) = g(x) = 0 uniformly with respect to x ∈ [0, b].

n→∞

So the proof follows from Theorem 4.1 immediately.

¤

References [1] P.N. Agrawal and H.S. Kasana, On simultaneous approximation by Sz´ asz-Mirakjan operators, Bull. Inst. Math. Acad. Sinica, 22, 181-188 (1994). [2] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Application, Walter de Gruyter Studies in Math. 17, de Gruyter & Co., Berlin, 1994. ¨ [3] O. Duman and M.A. Ozarslan, Sz´ asz-Mirakjan type operators providing a better error estimation, Appl. Math. Lett., (2006); in press (doi:10.1016/j.aml.2006.10.007). [4] Z. Finta, On converse approximation theorems, J. Math. Anal. Appl., 312, 159-180 (2005). [5] Z. Finta, N.K. Govil and V. Gupta, Some results on modified Sz´ asz-Mirakjan operators, J. Math. Anal. Appl., 327, 1284-1296 (2007). [6] V. Gupta, Simultaneous approximation for B´ezier variant of Sz´ asz-MirakyanDurrmeyer operators, J. Math. Anal. Appl., (in press). [7] V. Gupta and M.A. Noor, Convergence of derivatives for certain mixed Sz´ asz-Beta operators, J. Math. Anal. Appl., 321, 1-9 (2006). [8] V. Gupta, M.A. Noor and Beniwal, Rate of convergence in simultaneous approximation for Sz´ asz-Mirakyan-Durrmeyer operators, J. Math. Anal. Appl., 322, 964-970 (2006). [9] V. Gupta and R.P. Pant, Rate of convergence for the modified Sz´ asz-Mirakyan operators on functions of bounded variation, J. Math. Anal. Appl., 223, 476-483 (1999). [10] J.P. King, Positive linear operators which preserve x2 , Acta. Math. Hungar., 99, 203-208 (2003). [11] H.M. Srivastava and V. Gupta, A certain family of summation integral type operators, Math. Comput. Modelling, 37, 1307-1315 (2003). [12] V. Totik, Uniform approximation by Sz´ asz-Mirakjan type operators, Acta Math. Hungar., 41, 291-307 (1983). [13] X.-M. Zeng and A. Piriou, Rate of pointwise approximation for locally bounded functions by Sz´ asz operators, J. Math. Anal. Appl., 307, 433-443 (2005). TOBB Economics and Technology University, Faculty of Arts and Sci¨g ˘u ¨ to ¨ zu ¨ 06530, Ankara, Turkey ences, Department of Mathematics, So E-mail address: [email protected] Eastern Mediterranean University, Faculty of Arts and Sciences, Department of Mathematics, Gazimagusa, Mersin 10, Turkey E-mail address: [email protected] Eastern Mediterranean University, Faculty of Arts and Sciences, Department of Mathematics, Gazimagusa, Mersin 10, Turkey E-mail address: [email protected]

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SOME THEOREMS ON IF -COMPACT LINEAR OPERATORS HAKAN EFE Abstract. The purpose of this paper to introduce IF -compact operators in IF -normed linear spaces in the sense of Lael and Nourouzi [7]. Also classical compact operator by means of IF -concept have investigated.

1. Introduction The notion of fuzzy norm on a linear space was introduced by Katsaras [8] in 1984 …rstly. In 1992, Felbin [4] gave a de…nition of a fuzzy norm on a linear space whose associated metric is Kaleva type [5]. In 1994, Chang and Mordeson [3] introduced another idea of a fuzzy norm on a linear space whose associated metric is Kramosil and Michalek type [9]. Xiao and Zhu [11] rede…ned the idea of Felbin’s [4] de…nition of fuzzy norm of a linear operator from a fuzzy normed linear space to another fuzzy normed linear space. In 2003, Bag and Samanta [1] introduced a de…nition of a fuzzy norm and proved a decomposition theorem of a fuzzy norm into a family of crisp norms. In 2005, Bag and Samanta [2] gave an idea of fuzzy norm of a linear operator from a fuzzy normed linear space to another fuzzy normed linear space. They also de…ned various notion of continuities operators and boundedness of linear operators over fuzzy normed linear spaces. In 2004, Park [10] using the idea of intuitionistic fuzzy sets, de…ned the notion of intuitionistic fuzzy metric spaces with the help of continuous t-norm and continuous t-conorm as a generalization of fuzzy metric space. In 2006, Lael and Nourouzi [6] introduced fuzzy compact operators between fuzzy normed spaces. Very recently Lael and Nourouzi [7] gave a new de…nition for IF -normed linear space and proved some theorems: open mapping, closed graph and uniform boundedness in IF -normed linear spaces. In this paper, we introduce IF -compact operators in IF -normed linear spaces in the sense of Lael and Nourouzi. We have investigated classical compact operator by means of IF -concept. 2. Preliminaries De…nition 1 ([7]). The 3-tuple (X; ; ) is said to be an IF -normed linear space if X is a real vector space, and , are F -sets of X R satisfying the following conditions for every x; y 2 X and t; s 2 R, (i) (x; t) = 0, for all non-positive real number t, (ii) (x; t) = 1 for all t 2 R+ if and only if x = 0, t (iii) (cx; t) = (x; jcj ), for all t 2 R+ and c 6= 0, (iv) (x + y; s + t) minf (x; t); (y; s)g, (v) limt!1 (x; t) = 1 and limt!0 (x; t) = 0, 2000 Mathematics Subject Classi…cation. 46S40, 47A30, 47B07. Key words and phrases. IF -normed linear space, IF -compact operator. 1

62

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HAKAN EFE

(vi) (x; t) = 1, for all non-positive real number t, (vii) (x; t) = 0 for all t 2 R+ if and only if x = 0, t (viii) (cx; t) = (x; jcj ), for all t 2 R+ and c 6= 0, (ix) (x + y; s + t) maxf (x; t); (y; s)g, (x) limt!1 (x; t) = 0 and limt!0 (x; t) = 1. In this case, we will call ( ; ) an IF -norm on X. In addition, (X; ) is called an F -normed space. It is easy to see that for every x 2 X, the functions (x; ) and (x; ) are nondecreasing and nonincreasing on R, respectively. Lemma 1 ([7]). Let (X; ) be an F -normed linear space and (x; t) = 1 for all x 2 X and t 2 R. Then (X; ; ) is an IF -normed linear space. Example 1. Let (X; jj jj) be a normed space and that on X R de…ned by ( t if t > 0 t+jjxjj and 0 (x; t) = 0 (x; t) = 0 if t 0 1 (x; t) =

Then (

0;

(

jjxjj t

exp

0)

if if

0 and (

1;

1)

t>0 t 0

and

1 (x; t) =

(

0;

0;

jjxjj t+jjxjj

if if

1 1

and

1

1

1

be F -sets

t>0 , t 0 jjxjj t

exp

(x; t)

if if

t>0 : t 0

ae two IF -norms on X.

De…nition 2 ([7]). Let (X; ; ) be an IF -normed linear space and (xn )n be a sequence in X. Then (xn )n is said to be convergent to x 2 X if limn!1 (xn x; t) = 1 and limn!1 (xn x; t) = 0, for all t > 0. We denote it by xn ! x. De…nition 3. Let (X; ; ) be an IF -normed linear space and (xn )n be a sequence in X. Then (xn )n is said to be a Cauchy sequence, if limn!1 (xn+p xn ; t) = 1 and limn!1 (xn+p xn ; t) = 0 for all t > 0, p 2 N. Theorem 1 ([7]). Let (X; ; ) be an IF -normed linear space. that (F)

Assume further

(x; t) = 0 for all t > 0 implies x = 0.

De…ne jjxjj = ^ft > 0 : (x; t)

, (x; t)

1

g,

where 2 (0; 1). Then fjj jj : 2 (0; 1)g is an ascending family of norms on X, and they are called -norms on X corresponding to (or induced by) the IF -norm ( ; ) on X. Remark 1. Let (X; ; ) be an IF -normed linear space. Assume further that, for x 6= 0, (x; ) and (x; ) are continuous functions of R and is strictly increasing on ft > 0 : 0 < (x; t) < 1g and is strictly decreasing on ft > 0 : 0 < (x; t) < 1g. Let we show this (FF). Lemma 2 ([7]). Let (X; ; ) be an IF -normed linear space satisfying (F) and (xn )n be a sequence in X. Then xn ! x i¤ limn!1 jjxn xjj = 0 for all 2 (0; 1).

63

IF -NORM ED LINEAR SPACES

3

3. Main Results De…nition 4. Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces and T : X ! Y be a linear operator. The operator T is called IF -continuous at z 2 X if for any " > 0, and 2 (0; 1) there exist > 0 and 2 (0; 1) such that for all x 2 X, if 1 (x z; ) > and 1 (x z; ) < 1 then 2 (T (x) T (z); ") > and 2 (T (x) T (z); ") < 1 . If T is IF -continuous at each point of X, then T is said to be IF -continuous on X. De…nition 5. Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces and T : X ! Y be a linear operator. The operator T is called strongly IF -continuous at z 2 X if for any " > 0, there exist > 0 such that for all x 2 X, 2 (T (x) T (z); ") z; ) and 2 (T (x) T (z); ") z; ). 1 (x 1 (x If T is strongly IF -continuous at each point of X, then T is said to be strongly IF -continuous on X. De…nition 6. Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces and T : X ! Y be a linear operator. The operator T is called weakly IF -continuous at z 2 X if for any " > 0, and 2 (0; 1) there exist > 0 such that for all x 2 X, if 1 (x z; ) and 1 (x z; ) 1 then 2 (T (x) T (z); ") and T (z); ") 1 . 2 (T (x) If T is weakly IF -continuous at each point of X, then T is said to be weakly IF -continuous on X. De…nition 7. Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces and T : X ! Y be a linear operator. The operator T is called sequentially IF continuous at x 2 X if for any sequence (xn )n , xn 2 X for all n, with xn ! x implies T (xn ) ! T (x). I.e., for all t > 0, if limn!1 1 (xn x; t) = 1 and limn!1 1 (xn x; t) = 0 then limn!1 2 (T (xn ) T (x); t) = 1 and limn!1 2 (T (xn ) T (x); t) = 0. If T is sequentially IF -continuous at each point of X, then T is said to be sequentially IF -continuous on X. Remark 2. If a linear operator is strongly IF -continuous then it is weakly IF continuous. Theorem 2. Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces and T : X ! Y be a linear operator. If T is strongly IF -continuous then it is sequentially IF -continuous but not conversely. Proof. We assume that T is strongly IF -continuous at z 2 X. Then for any " > 0, there exist > 0 such that for all x 2 X, ((i))

2 (T (x)

T (z); ")

1 (x

z; ) and

2 (T (x)

T (z); ")

Let (xn )n be a sequence in X such that xn ! z, i.e., ((ii))

lim

n!1

1 (xn

z; t) = 1 and lim

n!1

1 (xn

z; t) = 0

1 (x

z; ).

64

4

HAKAN EFE

for all t > 0. Now from (i), 2 (T (xn ) T (z); ") 1 (xn T (z); ") z; ) for n = 1; 2; :::. This implies, 1 (xn lim

2 (T (xn )

T (z); ")

lim

2 (T (xn )

T (z); ")

n!1 n!1

z; ) and

lim

1 (xn

z; ) and

lim

1 (xn

z; ).

n!1 n!1

2 (T (xn )

Hence, limn!1 2 (T (xn ) T (z); ") = 1 and limn!1 2 (T (xn ) T (z); ") = 0 by (ii). Since " is a small arbitrary number, it follows that T (xn ) ! T (z). To show that sequential IF -continuity of T does not imply strong IF -continuity of T , consider the following example: Example 2. Let (X = R; jj jj) be a normed linear space where jjxjj = jxj for all x 2 R. De…ne 1 , 2 , 1 and 2 : X R ! [0; 1] by ( jxj t if t > 0 if t > 0 t+jxj t+jxj and 1 (x; t) = , 1 (x; t) = 0 if t 0 1 if t 0 ( jxj t if t > 0 if t > 0 t+kjxj t+kjxj , (x; t) = and (x; t) = 2 2 0 if t 0 1 if t 0 where k > 0 is a constant. Then (X; 1 ; 1 ) and (Y; 2 ; 2 ) are IF -normed linx4 ear spaces. If we consider the function T (x) = 1+x 2 , then T is sequentially IF continuous but not strongly IF -continuous. Theorem 3. Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces and T : X ! Y be a linear operator. Then T is IF -continuous i¤ it is sequentially IF -continuous. Proof. Suppose T is IF -continuous at z 2 X. Let (xn )n be a sequence in X such that xn ! z. Let " > 0 be given and choose 2 (0; 1). Since T is IF continuous at z 2 X, then there exist > 0 and 2 (0; 1) such that for all x 2 X, if 1 (x z; ) > and 1 (x z; ) < 1 then 2 (T (x) T (z); ") > and T (z); ") < 1 . Since xn ! z in X, there exists pozitive integer n0 2 (T (x) such that 1 (xn z; ) > and 1 (xn z; ) < 1 for all n n0 . Then T (z); ") > and 2 (T (xn ) T (z); ") < 1 for all n n0 . So 2 (T (xn ) for a given " > 0 and for any 2 (0; 1), there exists pozitive integer n0 such that and 2 (T (xn ) T (z); ") < 1 for all n n0 . This implies 2 (T (xn ) T (z); ") > limn!1 2 (T (xn ) T (z); ") = 1 and limn!1 2 (T (xn ) T (z); ") = 0. Since " > 0 is arbitrary, thus T (xn ) ! T (z) in (Y; 2 ; 2 ). Next we suppose that T is sequentially IF -continuous at z 2 X. If possible assume that T is not IF -continuous at z 2 X. Thus there exists " > 0 and 2 (0; 1) such that for any > 0 and 2 (0; 1), there exists y(depending on , ) such that 1 (z 2 (T (z)

y; ) > T (y); ")

and and

1 (z

y; ) < 1 but (T (z) T (y); ") 1 2

(i) .

1 1 Thus for = 1 n+1 , = n+1 , n = 1; 2; :::, there exists yn such that 1 (z 1 1 1 1 yn ; n+1 ) > 1 n+1 and 1 (z yn ; n+1 ) < n+1 but 2 (T (z) T (yn ); ") and 1 (T (z) T (y ); ") 1 . Taking > 0, there exists n such that < for 2 n 0 n+1 1 1 all n n0 . Then 1 (z yn ; ) yn ; n+1 ) > 1 n+1 and 1 (z yn ; ) 1 (z

65

IF -NORM ED LINEAR SPACES

5

1 1 yn ; n+1 ) < n+1 for all n n0 . This implies limn!1 1 (z yn ; ) 1 and limn!1 1 (z yn ; ) 0. Hence yn ! z. But from (i), 2 (T (z) T (yn ); ") and (T (z) T (y ); ") 1 so (T (z) T (y ); ") 9 1 and (T (z) T (y ); ") 9 0 2 n n 2 n 2 as n ! 1. Thus T (yn ) does not convergence to T (z) whereas yn ! z (w.r.t. ( 1 ; 1 )), which is a contradiction to our assuption. Hence T is IF -continuous at z. 1 (z

De…nition 8 ([7]). Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces and T : X ! Y be a linear operator. The operator T is called weakly IF -bounded if for any 2 (0; 1), there exists constant h > 0 such that for every x 2 X and " > 0, and 1 (h x; ") 1 ) 2 (T (x); ") and 2 (T (x); ") 1 . 1 (h x; ") De…nition 9 ([7]). Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces and T : X ! Y be a linear operator. The operator T is called strongly IF -bounded if there exists constant h > 0 such that for every x 2 X and " > 0, 2 (T (x); ") 1 (hx; "). 1 (hx; ") and 2 (T (x); ") Theorem 4 ([7]). Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces satisfying (F), where 2 (x; ) and 2 (x; ) are continuous function on R for all x 2 X. If the linear operator T : (X; 1 ; 1 ) ! (Y; 2 ; 2 ) is bounded w.r.t. norms of ( 1 ; 1 ) and ( 2 ; 2 ), then T is weakly IF -bounded. Theorem 5. Let (X; 1 ; 1 ) and (Y; 2 ; T : X ! Y be a linear operator. Then

2)

be two IF -normed linear spaces and

(i) T is strongly IF -continuous on X i¤ T is strongly IF -continuous at a point x0 2 X. (ii) T is strongly IF -continuous i¤ T is strongly IF -bounded. Proof. (i) Since T is strongly IF -continuous at x0 2 X, thus for each " > 0, there exist > 0 such that for all x 2 X, 2 (T (x) T (x0 ); ") x0 ; ) and 1 (x 2 (T (x) T (x0 ); ") 1 (x x0 ; ). Taking any y 2 X and replacing x by x + x0 y we get 2 (T (x

+ x0

y)

T (x0 ); ")

1 (x

) )

+ x0 y x0 ; ) (T (x) + T (x0 ) T (y) T (x0 ); ") 2 T (y); ") y; ), 2 (T (x) 1 (x

1 (x

y; )

1 (x

y; )

and 2 (T (x

+ x0

y)

T (x0 ); ")

1 (x

) )

+ x0 y x0 ; ) T (y) T (x0 ); ") 2 (T (x) + T (x0 ) T (y); ") y; ). 2 (T (x) 1 (x

Since y 2 X is arbitrary, it follows that T is strongly IF -continuous on X. (ii) We suppose that T is strongly IF -continuous. Using continuity of T at x = 0, for " = 1, there exists > 0 such that 2 (T (x) T (0); 1) 0; ) and 1 (x (T (x) T (0); ") (x 0; ), for all x 2 X. 2 1 Suppose that x 6= 0 and t > 0. Putting u = x=t then 2 (T (x); t)

=

2 (tT (u); t)

=

2 (T (u); 1)

1 (u;

)=

1

x ; t

=

1

x;

t h

66

6

HAKAN EFE

and 2 (T (x); t)

=

2 (tT (u); t)

=

2 (T (u); 1)

1 (u;

)=

1

x ; t

=

1

x;

t h

where h = 1= . So 2 (T (x); t) 1 (hx; t). 1 (hx; t) and 2 (T (x); t) If x 6= 0 and t 0 then 2 (T (x); t) = 0 = 1 (hx; t) and 2 (T (x); t) = 0 = 1 (hx; t). If x = 0 and t 2 R then T (0) = 0 and 2 (0; t)

=

1

2 (0; t)

=

1

t h t 0; h 0;

= 1 and

2 (0; t)

=

1

= 0 and

2 (0; t)

=

1

t h t 0; h 0;

= 0 if t > 0, = 1 if t

0.

Hence T is strongly IF -bounded. Conversely suppose that T is strongly IF -bounded. Then there exists h > 0 such that 2 (T (x); ") 1 (hx; ") for all x 2 X and for 1 (hx; ") and 2 (T (x); ") all " > 0. This implies " " 0; ) T (0); ") 0; ) and 2 (T (x) T (0); ") 1 (x 2 (T (x) 1 (x h h for all x 2 X and for all " > 0. Hence 2 (T (x)

T (0); ")

1 (x

0; ) and

2 (T (x)

T (0); ")

1 (x

0; )

where = h" . This implies that T is strongly IF -continuous at 0 and hence it is strongly IF -continuous on X. Theorem 6. Let (X; 1 ; 1 ) and (Y; 2 ; T : X ! Y be a linear operator. Then

2)

be two IF -normed linear spaces and

(i) T is weakly IF -continuous on X i¤ T is weakly IF -continuous at a point x0 2 X. (ii) T is weakly IF -continuous i¤ T is weakly IF -bounded. Proof. (i) Since T is weakly IF -continuous at x0 2 X, thus for each " > 0, there exist > 0 such that for all x 2 X, 1 (x x0 ; ) and 1 (x x0 ; ) 1 ) (T (x) T (x ); ") and (T (x) T (x ); ") 1 . Taking any y 2 X and 0 2 0 2 replacing x by x + x0 y we get 1 (x

)

2 (T (x

+ x0

+ x0

y)

y

x0 ; ) T (x0 ); ")

and and

1 (x

+ x0

2 (T (x

y

+ x0

x0 ; ) y)

1

T (x0 ); ")

1

.

I.e.,

1 (x

y; ) )

and 1 (x y; ) T (y); ") 2 (T (x)

1 and

2 (T (x)

T (y); ")

1

.

Since y 2 X is arbitrary, it follows that T is weakly IF -continuous on X. (ii) We suppose that T is weakly IF -continuous. Using continuity of T at x = 0, for " = 1, there exists > 0 such that 1 (x 0; ) and 1 (x 0; ) 1 implies 2 (T (x) T (0); 1) and 2 (T (x) T (0); 1) 1 , for all x 2 X. I.e., (x; ) and (x; ) 1 implies (T (x); 1) and 1 , 1 2 (T (x); 1) 1 2 for all x 2 X.

67

IF -NORM ED LINEAR SPACES

7

Suppose that x 6= 0 and t > 0. Putting x = u=t then 1 (u=t;

1 (u; t

) )

)

)

1 (h u; t)

)

1

and 1 (u=t; ) 1 and 2 (T (u=t); 1) 2 (T (u=t); 1) and 1 (u; t ) 1 and 2 (T (u); t) 1 2 (T (u); t) and 1 (h u; t) 1 and 2 (T (u); t) 1 2 (T (u); t)

1

, i.e.,

, i.e.,

where h = . This implies T is weakly IF -bounded. If x 6= 0 and t 0 then 1 (h x; t) = 2 (T (x); t) = 0 and (T (x); t) = 1 for any h > 0. 2 If x = 0 and t 2 R then for h > 0, (h 1 (h

1

0; t) = 0; t) =

2 (T (0); t)

= 1 and (T (0); t) = 0 and 2

(h 1 (h

1

0; t) = 0; t) =

1

(h x; t) =

2 (T (0); t)

= 0 if t > 0, (T (0); t) = 1 if t 0. 2

Hence T is weakly IF -bounded. Conversely suppose that T is weakly IF -bounded. Then there exists h > 0 such that 1 (h x; t) and 1 (h x; t) 1 ) 2 (T (x); t) and 2 (T (x); t) 1 for all x 2 X and for all t 2 R. This implies x

1

0;

t h

and )

t h T (0); t) x

1

2 (T (x)

1

0;

and

2 (T (x)

T (0); t)

1

2 (T (x)

T (0); ")

1

.

Then, x

1

0;

" h

and )

" h T (0); ") x

1

2 (T (x)

1

0;

and

for " > 0. Hence 1

(x

0; ) )

and 1 (x 0; ) T (0); ") 2 (T (x)

1 and

2 (T (x)

T (0); ")

1

where = h" . This implies that T is weakly IF -continuous at x = 0 and hence it is weakly IF -continuous on X. Lemma 3. Let (X; ; ) be an IF -normed linear spaces satisfying (F) and (FF) and fjj jj : 2 (0; 1)g be the family of corresponding -norms of ( ; ) on X. Then for x0 2 X, x0 6= 0, (x0 ; jjx0 jj ) and (x0 ; jjx0 jj ) 1 for all 2 (0; 1). Proof. Let jjx0 jj = , then > 0. There exists a sequence (tn )n , tn > 0, n 2 N such that (x0 ; tn ) , (x0 ; tn ) 1 and tn # . Therefore, lim

n!1

(x0 ; tn )

and lim

n!1

(x0 ; tn )

)

(x0 ; lim tn )

)

(x0 ; jjx0 jj )

n!1

1

and (x0 ; lim tn )

1

n!1

and (x0 ; jjx0 jj )

1

by (FF) forall

2 (0; 1).

68

8

HAKAN EFE

Lemma 4. Let (X; ; ) be an IF -normed linear spaces satisfying (F) and (FF) and fjj jj : 2 (0; 1)g be the family of corresponding -norms of ( ; ) on X. Then for x0 6= 0, 2 (0; 1) and t0 > 0, jjx0 jj = t0 i¤ (x0 ; t0 ) = and (x0 ; t0 ) = 1 . Proof. Let 2 (0; 1), x0 6= 0 and t0 = jjx0 jj = ^fs : (x0 ; s) , (x0 ; s) 1 g. Since (x; ) and (x; ) are continuous (by (FF)), from Lemma 3 we have (x0 ; t0 ) and (x0 ; t0 ) 1 . Also (x0 ; t0 ) (x0 ; s) if (x0 ; s) 0 and (x0 ; t ) (x0 ; s) if (x0 ; s) 1 . If possible, let (x0 ; t0 ) > and (x0 ; t0 ) < 1 , then by the continuity of (x0 ; ) and (x0 ; ) at t0 , there exists t00 < t0 such that (x0 ; t00 ) > and (x0 ; t00 ) < 1 which is impossible, since t0 = ^fs : (x0 ; s) , (x0 ; s) 1 g. Thus (x0 ; t0 ) and (x0 ; t0 ) 1 . 0 0 Hence we get (x0 ; t ) = and (x0 ; t ) = 1 . Next if (x0 ; t0 ) = and (x0 ; t0 ) = 1 , 2 (0; 1), then from the de…nition jjx0 jj = ^ft : (x0 ; t) , (x0 ; t) 1 g = t0 (Since (x0 ; ) is strictly increasing on ft > 0 : 0 < (x; t) < 1g and is strictly decreasing on ft > 0 : 0 < (x; t) < 1g). This completes the proof. Theorem 7. Let (X; ; ) be an IF -normed linear spaces satisfying (F) and (FF) and fjj jj : 2 (0; 1)g be the family of corresponding -norms of ( ; ) on X. Then for any increasing (decreasing) sequence ( n )n in (0; 1), n ! implies jjxjj n ! jjxjj for all x 2 X. Proof. For x = 0, clearly n ! implies jjxjj n ! jjxjj . Suppose x 6= 0. From Lemma 4, for x 6= 0, 2 (0; 1) and t0 > 0 we have jjxjj = 0 t i¤ (x; t0 ) = and (x; t0 ) = 1 . Let ( n )n be an increasing sequence in (0; 1) such that n ! 2 (0; 1). Let jjxjj n ! tn and (x; t) = and (x; t) = 1 . Since fjj jj : 2 (0; 1)g is an increasing family of norms, (tn )n is an increasing sequence of real numbers and it is bounded above by t (since jjxjj n jjxjj for all n 2 N). Hence (tn )n is convergent. Thus limn!1 (x; tn ) = limn!1 n and limn!1 (x; tn ) = limn!1 (1 and n ) which implies (x; limn!1 tn ) = (x; limn!1 ; tn ) = 1 . Thus we have (x; limn!1 tn ) = (x; t) and (x; limn!1 ; tn ) = (x; t), which implies limn!1 tn = t by (FF). Therefore limn!1 jjxjj n = jjxjj . Similarly, if ( n )n is a decreasing sequence in (0; 1) and n ! 2 (0; 1), then it can be shown that jjxjj n ! jjxjj for all x 2 X. Theorem 8. Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces satisfying (F) and (FF) and T : X ! Y be a linear operator . Then T is weakly IF -bounded i¤ T is bounded w.r.t. -norms of ( 1 ; 1 ) and ( 2 ; 2 ), 2 (0; 1). Proof. First we suppose that T is weakly fuzzy bounded. Thus for all there exists h > 0 such that for all x 2 X, for all t 2 R we have 1

(h x; t)

and

1

(h x; t)

1

)

2 (T (x); t)

and

2 (T (x); t)

2 (0; 1), 1

.

I.e. _f

_f

2 (0; 1) : jjh xjj1

2

2 (0; 1) : jjT (x)jj

tg tg

)

(i)

.

Now we show that, _f 2 (0; 1) : jjh xjj1

tg

, jjh xjj1

t.

69

IF -NORM ED LINEAR SPACES

9

If x = 0 then the relation is obvious. Suppose x 6= 0. Now if _f 2 (0; 1) : jjh xjj1

(ii)

then jjh xjj1

tg >

t.

If _f 2 (0; 1) : jjh xjj1 tg = , then there exists an increasing sequence ( in (0; 1) such that n " and jjh xjj1 n t. Then by Theorem 7 we have jjh xjj1

(iii)

n )n

t.

Thus from (ii) and (iii) we get, _f 2 (0; 1) : jjh xjj1

(iv)

Next we suppose that jjh xjj1 t. If jjh xjj1 < t then 1 (h x; t)

) jjh xjj1

tg

and

1

(h x; t)

_f 2 (0; 1) : jjh xjj1

(v) 1

tg

1

t. . I.e.

.

If jjh xjj = t i.e. ^fs : 1 (h x; s) and 1 (h x; s) 1 exists a decreasing sequence (sn )n in R such that sn # t and 1 1 (h x; sn ) ) limn!1 1 (h x; sn ) and limn!1 1 (h x; sn ) 1 ) 1 (h x; limn!1 sn ) and 1 (h x; limn!1 sn ) 1 ) 1 (h x; t) and 1 (h x; t) 1 . ) _f 2 (0; 1) : jjh xjj1

(vi)

tg

g = t, then there (h x; sn ) and 1 by (FF).

.

From (v) and (vi) it follows that, jjh xjj1

(vii)

t ) _f 2 (0; 1) : jjh xjj1

tg

.

tg

, jjh xjj1

t.

tg

, jjT (x)jj2

Hence from (iv) and (vii) we have, _f 2 (0; 1) : jjh xjj1

(viii)

In a similar way we can show that, _f 2 (0; 1) : jjT (x)jj2

(ix)

t.

Therefore from (viii) and (ix) we have 1

(h x; t)

and

1

(h x; t)

1

)

2

(T (x); t)

and

2

(T (x); t)

1

t ) jjT (x)jj2 t. This implies that jjT (x)jj2 h jjxjj1 for all then jjh xjj1 2 (0; 1). Conversely suppose that for all 2 (0; 1), there exists h > 0 such that jjT (x)jj2 h jjxjj1 for all x 2 X. Then for x 6= 0, jjh xjj1 t ) jjT (x)jj2 t, for all t > 0, i.e., ^fs : ^fs :

(h x; s) 2 (T (x); s)

1

and and

(h x; s) 1 1 2 (T (x); s)

g

1

t) g t.

In a similar way as above we can show that ^fs

: ,

(h x; s) 1 (h x; s)

and and

(T (x); s) 2 (T (x); t)

and and

1

(h x; s) 1 (h x; s)

1 1

g

(T (x); s) 2 (T (x); t)

1 1

g .

1

t

and ^fs

: ,

2

2

t

70

10

HAKAN EFE

Thus we have 1

(h x; t)

and

1

(h x; t)

1

)

2

(T (x); t)

and

2

(T (x); t)

1

for all x 2 X. If x 6= 0, t 0 and x = 0, t > 0 then the above relation is obvious. Hence the theorem follows. 4. IF -Compact Operators De…nition 10. A subset A of an IF -normed linear space (X; ; ) is said to be IF -bounded i¤ there exist t > 0 and 0 < r < 1 such that (x; t) > 1 r and (x; r) < r for all x 2 A. De…nition 11. A subset A of an IF -normed linear space (X; ; ) is said to be IF -compact if any sequence (xn )n in A has a subsequence converging to an element of A. De…nition 12. The IF -closure of a subset B of an IF -normed linear space (X; ; ) is denoted by B and de…ned by the set of all x 2 X such that there is a sequence (xn )n of elements of B with xn ! x. We say that B is IF -closed if B = B. De…nition 13. Let (X; 1 ; 1 ) and (Y; 2 ; 2 ) be two IF -normed linear spaces. A linear operator T : X ! Y is called IF -compact operator if for every IF -bounded subset M of X the subset of T (M ) of Y is relatively compact, that is the IF -closure of T (M ) is a IF -compact set. Example 3. Let (X; jj jj1 ) and (Y; jj jj2 ) be two ordinary normed linear spaces, and T : X ! Y be a compact operator. Then it is easy to see that T : (X; 1 ; 1 ) ! (Y; 2 ; 2 ) is a IF -compact operator, where ( 1 ; 1 ) and ( 2 ; 2 ) are the standard IF -norms induced by ordinary norms jj jj1 and jj jj2 , respectively, i.e., i (x; t)

and i (x; t)

t t+jjxjji

=

=

0 (

jjxjji t+jjxjji

0

if if

t > 0, t 2 R, , t 0

if if

t > 0, t 2 R, t 0

for i = 1; 2. Example 4. Let C[0; 1] be the set of all real valued continuous functions on [0; 1] with the IF -norm supx2[0;1] j'(x)j t and ('(x); t) = , ('(x); t) = t + supx2[0;1] j'(x)j t + supx2[0;1] j'(x)j where '(x) 2 C[0; 1] and t > 0. If k(x; y) with x; y 2 [0; 1] is a real valued continuous function, then the operator T : C[0; 1] ! C[0; 1] de…ned by Z 1 (T ')(x) = k(x; y)'(y)dy, 0

where ' 2 C[0; 1] is an IF -compact operator.

Theorem 9. Let T : (X; 1 ; 1 ) ! (Y; 2 ; 2 ) be a linear operator. Then T is IF -compact i¤ it maps every IF -bounded sequence (xn )n in X onto a sequence (T (xn ))n in Y which has an IF -convergent subsequence.

71

IF -NORM ED LINEAR SPACES

11

Proof. Suppose that T be a IF -compact operator and (xn )n be an IF -bounded sequence in (X; 1 ; 1 ). The IF -closure of fT (xn ) : n 2 Ng is an IF -compact set. So (T (xn ))n has an IF -convergent subsequence by de…nition. Conversely, let A be a IF -bounded subset of (X; 1 ; 1 ). We show that the IF -closure of T (A) is IF -compact. Let (xn )n be a sequence in the closure of T (A). For given " > 0, n 2 N and t > 0, there exists (yn )n in T (A) such that 2 (xn yn ; 2t ) > 1 " and yn ; 2t ) < ". Let yn = T (zn ), where zn 2 A. Since A is IF -bounded set, so 2 (xn is fzn : n 2 Ng. On the other hand, because T is IF -compact operator, T (zn ) has an IF -convergent subsequence (ynk )k = (T (znk ))k . Let ynk ! y for some y 2 Y . Hence 2 ynk y; 2t > 1 " and 2 ynk y; 2t < " for all nk > n0 . We have 2

(xnk

y; t)

min

2

xnk

2

(xnk

y; t)

max

2

xnk

t ; 2 t ; y nk ; 2

y nk ;

2

y nk

2

y nk

t 2 t y; 2

y;

>1

",

0 the set fk 2 N : jxk Lj "g has the natural density zero. A generalization of statistical convergence which is based on the structure of the ideal I of subsets of N is given by Kostyrko, Maµcaj, Šalát and Sleziak [6]. A non-void class I P (N) is called the ideal if I is additive ( i.e., A, B 2 I ) A [ B 2 I) and hereditary (i.e., A 2 I, B A ) B 2 I). Throughout in this paper we consider admissible ideals, i.e. those which are di¤erent from P (N) and contain all singletons. It is easy to check that I = fK N : d (K) = 0g forms an admissible ideal. A sequence x = (xk ) of real numbers is I-convergent to L if fk : jxk Lj "g 2 I for every " > 0. In this case we write I lim x = L. It is known that any convergent sequence is I-convergent, but not conversely. Some examples and properties of I-convergence may be found in [6]. 1 2 2. The Main Results For a sequence (Ln ) of positive linear operators on C (a; b), which is the space of continuous functions on [a; b] and bounded on real axis R, Korovkin [5] solved a problem which based on the existence of the limit limn Ln (f ; x) = f (x). Also, Curtis [1] has extended this theorem for the functions in Lp ( ; ). Gadjiev [3] stated and proved weighted Korovkin type theorems in the space of locally integrable functions in R. Recently, Gadjiev and Aral [4] have investigated Korovkin type approximation theorems in the weighted spaces Lp; ! (R) and Lp; (Rn ). The purpose of the present paper is to study a Korovkin type approximation theorem via I-convergence in the weighted space Lp; ! (R) and Lp; (Rn ). 1 2000 AMS Subject Classi…cations: Primary 41A10, 41A25, 41A36; Secondary 40A05, 40A30. Key words and phrases. I-convergence, positive linear operator, Korovkin theorem, weighted space. 2 Corresponding author Email Address : [email protected] (F. Dirik), [email protected] (A. Aral), [email protected] (K. Demirci)

1

76

DIRIK ET AL

I-CONVERGENCE OF POSITIVE LINEAR OPERATORS ON Lp WEIGHTED SPACES2

For a …xed p 2 [1; 1); let ! be a positive continuous function on R satisfying the condition Z t2p ! (t) dt < 1. (2.1) R

By Lp; ! (R) (1 p < 1) we denote the linear space of all functions f which are measurable, p absolutely integrable on R with respect to the weight function !, that is, 9 8 0 1 p1 > > Z = < p Lp; ! (R) = f : R ! R : kf kp; ! := @ jf (t)j ! (t) dtA < 1 . > > ; : R

The minimum and maximum values of the function ! on …nite intervals will be denoted by ! min and ! max respectively.

Theorem 1. Let I be an admissible ideal in N. Let (Ln )n2N be the sequence of positive linear operators Ln : Lp; ! (R) ! Lp; ! (R) and let the sequence fkLn kg be a uniformly bounded. If I lim Ln ti ; x xi p; ! = 0; i = 0; 1; 2, (2.2) then for any function f 2 Lp; ! (R) ; we have I

lim kLn f

f kp;

!

= 0:

Proof. We follow the proof of Theorem 1 in [4] up to a certain stage. We can choose a such large number A such that for every " > 0 A 2 p; !

f

< ",

(2.3)

where A 1 be characteristic function of the interval [ A; A] and Since Ln is linear operator, we have kLn f

f kp; !

=

Ln

A 1

Ln

A 1f

0

A 2

+

A 1

f A 1 f p; !

+

A 2

f

+ Ln

A 2

p; ! A 2f

(t) = 1

A 1

(t). (2.4)

A 2 f p; !

00

= In + In . 00

Firstly, we compute In : Since fLn g is a uniformly bounded sequence, there exists a constant K > 0 such that kLn kp; ! K. (2.5) Hence, from (2:3), we have 00

In

Ln

A 2f

(K + 1)

+

p; ! A 2 f p; !

A 2 f p; !

< (K + 1) ": Furthermore, for every function f 2 Lp;! (R) the inequality k

A 1 f kp

1=p

! min kf kp;!

(2.6)

77

I-CONVERGENCE OF POSITIVE LINEAR OPERATORS ON Lp WEIGHTED SPACES3

implies Lp;! (R) Lp ( A; A). Since the space of continuous functions is dense in Lp ( A; A), given f 2 Lp;! (R), for each "0 > 0; there exists a continuous function ' on [ A; A] satisfying the condition '(x) = 0 for jxj > A such that (f

A 1 p

')

"0


0 such that

.

Thus, Ln '

A 1

'

A1 1

A 1

Ln

p; !

A1 1

+ ' (x) "0 + +

2M' 2

4M' 2

A 1

' (t)

(t)

' (x)

A 1

(x) (Ln (1; x)

1)

xkp; ! +

(x) (2.11) p; !

p; !

A2 + M' kLn (1; x)

A kLn (t; x)

A1 1

(x) ; x

1kp;!

2M'

Ln t2 ; x

2

x2 .

Substituting (2:11) into (2:10), we get In0

2"0 + "0 + +

2M' 2

2M' 2

A2 + M' kLn (1; x)

Ln t2 ; x

x2

p; !

4M'

1kp;! +

2

A kLn (t; x)

xkp; !

.

Then, the inequality (2:4) is obtained the following, kLn f

2"0 + (K + 1) " + "0 +

f kp; !

+

4M' 2

A kLn (t; x)

2M' 2

A2 + M' kLn (1; x)

xkp; ! +

2M' 2

Ln t2 ; x

1kp;!

x2

p; !

.

We conclude that kLn f

f kp; !

n where B := max "0 +

n 2"0 + (K + 1) " + B kLn (1; x) o x2 p; ! + Ln t2 ; x 2M' 2

A2 + M' ;

00

" > 0, n n : kLn f

f kp; !

"

00

o

4M' 2

;

2M' 2

1kp;! + kLn (t; x)

xkp; !

o . The last inequality shows that, for any

fn : 2"0 + (K + 1) " + B fkLn (1; x) + kLn (t; x) + Ln t2 ; x

xkp; ! x2

p; !

"

00

oo

.

1kp;! (2.12)

79

I-CONVERGENCE OF POSITIVE LINEAR OPERATORS ON Lp WEIGHTED SPACES5

Now we write D D1 D2 D3

: : : :

n = n : kLn f ( =

= =

f kp; !

n : kLn (1; x)

(

"

o

1kp; !

n : kLn (t; x)

xkp;!

n : Ln (t2 ; x)

x2

(

00

p;!

,

) (2"0 + (K + 1) ") , 3B ) 00 " (2"0 + (K + 1) ") , 3B ) 00 " (2"0 + (K + 1) ") . 3B "

00

Then, it follows from (2:12) that D D1 [ D2 [ D3 . By (2:2), Di 2 I for each i = 1; 2; 3. So, by the de…nition of ideal, D1 [ D2 [ D3 2 I, which yields D 2 I. So we have n o 00 n : kLn f f kp; ! " 2 I whence the result. Now we give an example of a sequence of positive linear operators such that this operator satis…es the conditions of Theorem 1 but does not satisfy the conditions of classical Korovkin theorem in weighted space Lp; ! (R). p

Example 2. We choose ! (x) = 1+x1 6m , p 1. By Lp; m (R) we denote the space of Lp; ! (R). Note that this selection of ! (x) satis…es the condition (2:1). Also note that for 1 p :

0

=@

Z

Rn

p

jf (t)j

11=p

(t) dtA

9 > =

;

.

Theorem 3. Let I be an admissible ideal in N. Let (Ln )n2N be the sequence of positive linear operators Ln : Lp; (Rn ) ! Lp; (Rn ) and let the sequence fkLn kg be a uniformly bounded. If I I I

lim

lim kLn (1; x)

1kp;

=

0,

xi kp;

=

0, i = 1; 2; ::; n,

=

0,

n!1

lim kLn (ti ; x)

n!1

n!1

2

Ln jtj ; x

2

jxj

p;

then for any function f 2 Lp; (Rn ),we have I Proof.

lim kLn f

n!1

f kp;

= 0.

Using a similar idea, the wanted result is obtained.

Conclusion 4. If we de…ne If = fK N : jKj < 1g, Id = fK N : d (K) = 0g, IdA = fK N : dA (K) = 0g and I = fK N : (K) = 0g, then we get the de…nitions of usual convergence, statistical convergence, A-statistical convergence and -statistical convergence, respectively. Details may be found in [5]. So, Theorem 1 and Theorem 3 are valid in this cases. References [1] P. C., Jr Curtis, The degree of approximation by positive convolution operators, Michigan Math. J., No:2, 12, 153-160 (1965) [2] H. Fast, Sur la convergence statistique, Colloq. Math., 2, 241-244, (1951). [3] A. D. Gadjiev, R. O. Efendiyev, E. Ibikli, On Korovkin type theorem in the space of locally integrable functions, Czechoslovak Mat. Journal 53, No:1, (128), 45-53, (2003). [4] A. D. Gadjiev and A. Aral, Weighted Lp -approximation with positive linear operators on unbounded sets, Appl. Math. Lett., doi:10.1016/j.aml.2006.12.007, (2007).

81

I-CONVERGENCE OF POSITIVE LINEAR OPERATORS ON Lp WEIGHTED SPACES7

[5] P. P. Korovkin, Linear operators and the theory of approximation, India, Delhi, (1960). [6] P. Kostryko, M. Maµcaj, T. Šalát, M. Sleziak, I-convergence and Extremal I-limit points, Math. Slovaca, 55, No. 4, 443-464, (2005). [7] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. math. 2, 73-74, (1951). [8] O. Szász, Generalization of S. Bernstein’s polynomials to in…nite interval, J. Research Nat. Bur. Standards, 45, 239-245, (1959).

82

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.1,83-100,2008, COPYRIGHT 2008 EUDOXUS PRESS, 83 LLC

Identi…cation of planar screens at low frequencies in thermoelasticity Drossos Gintides and Kiriakie Kiriaki Department of Mathematics National Technical University of Athens Zografou Campus, 15780 Athens, Greece Abstract In this paper the problem of determining a screen in an isotropic and homogeneous thermoelastic medium at low frequencies is considered. We formulate the direct problem for the planar screen in the thermoelastic medium and present an equivalent model for the problem under consideration at low-frequencies based on an non - homogeneous formulation via appropriate Dirac measures. We prove that the corresponding inverse problem of reconstructing the planar screen for two important cases: the thermal stress dislocation and the thermal displacement dislocation from boundary measurements has a unique solution. Finally, we present a reconstruction method for the above cases based on a proper use of certain vector test functions and the application of the two-sided Laplace transform.

1

Introduction

Scattering theory has played a central role in the scienti…c area of mathematical physics. The inverse scattering problem is a basic one in many applications, so, it concentrates the most interest and is in the foreground of the mathematical research. An excellent presentation of theoretical results and methods exploiting the inverse problem can be found in [10]. Problems connected with the scattering of waves by a very thin obstacle became very important, …nding applications especially in non-destructive tests. For the 1

84

GINTIDES-KIRIAKI

identi…cation of cracks by boundary measurements signi…cant results among others have been obtained by Friedman and Vogelius [14], Alessandrini et al [1] and Kress et al [9]. Amari et al in a series of papers, between them [2, 3], have proposed a new method for solving inverse problems using lowfrequency waves. More precisely in [2] a computational attractive method is introduced for the identi…cation of planar cracks located deep inside a heterogeneous conducting body based on low-frequency asymptotic analysis of Maxwell’s equations. In [4] complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion are presented while in [15] identi…cation of elastic inclusions and elastic moment tensors by boundary measurements is investigated. In this work we examine an inverse scattering problem for a screen in thermoelasticity. Thermoelasticity combines the theory of elastodynamics with the theory of heat conduction. The governing equations form a coupled system of three hyperbolic and one parabolic equation. The hyperbolic equations have a source term which is proportional to the temperature gradient, while the parabolic equation has a source term which is proportional to the divergence of the velocity. In [16] can be found an excellent introduction of the theory of thermoelasticity. In [11] the scattering process, the scattering amplitudes and cross sections in thermoelasticity are presented. The theory of thermoelastic waves in the low-frequency region has been developed in [11]. Results for speci…c applications at low frequencies can be found in [8, 12]. In [7] a complete analysis of the three-dimensional thermoelastic scattering problems from screens is presented. In a recent work [5] the reciprocity gap principle is exploited in order to identify planar cracks in which are all located in the same plane in a homogeneous and isotropic thermoelastic medium. Their method estimates the plane by an explicit formula. In Section 2 we formulate the direct scattering problem for the planar screen in the thermoelastic medium. Two types of screens are considered, thermal stress dislocations and displacements dislocations. The governing equations, the boundary conditions for each case and the radiation conditions for the well-posedness of the problem are presented. In Section 3 an equivalent model for the problem under consideration at low-frequencies is presented. Based on results for the elastic case given in [6] we prove that the direct problem for the screen can be replaced by a model prescribed by a nonhomogeneous equation having a force term corresponding to a thermoelastic sources distribution on the planar screen. In Section 4 we face the inverse problem to determine the shape of the screen and the estimation of the cor2

LOW FREQUENCES IN THERMOELASTICITY

85

responding Burgers vectors which describe mainly the physical properties of the dislocation [6]. The reconstruction method for the thermal stress dislocation as well as the thermal displacement dislocation is presented. Firstly, the method gives the plane of the screen and secondly provides a reconstruction of the screen by an inversion of a two-sided Laplace transform applied on data produced by Green’s formula.

2

Formulation of the direct scattering problem

We consider the scattering process in a homogeneous and isotropic thermoelastic medium of Biot type in R3 . The Biot medium is characterized by the real Lamé constants ; where > 0, + 2 > 0 and mass density , the coe¢ cient of thermal di¤usity and the coupling constants and . We assume that inside the thermoelastic medium we have a planar dislocation which is a bounded, simply connected, orientable smooth surface with a smooth non-self intersecting boundary. We denote the two sides of as + and and we will use the superscripts + and - to indicate that the corresponding quantities are measured at neighboring points on + and respectively. The unit normal vector is denoted as b and, due to the fact that we consider a planar dislocation, it is the same on each point of . We assume that b = b = b + . If we consider only time harmonic …elds then the Biot system assumes the following spectral form + ! 2 u (x) = r (x) ; x 2R3 n i! (x) = i ! r u (x) ; x 2R3 n +

(1) (2)

where u (x) is the elastic displacement, (x) denotes the thermal variation …eld, is the Lamé operator and ! is the frequency. We de…ne as q = i ! the spectral thermal constant. Note that whenever ! 0+ , ! 0+ the Biot system decouples into the Navier equation of dynamic elasticity and the heat conduction equation. Introducing a four dimensional vector notation as in [11], we have u (x) U (x) := (3) (x) 3

86

GINTIDES-KIRIAKI

and the Biot system (1, 2) is simply written as LU (x) = 0; ; x 2R3 n

(4)

+ ! 2 ) I3 r . The q r +q thermoelastic operator L is a 4x4 matrix elliptic di¤erential operator. It is not self-adjoint and the adjoint operator L may be obtained by L by replacing with i ! and vice versa. A uni…ed …eld U (x) which satis…es (4) admits a decomposition into three vector …elds

where L is the di¤erential operator L :=

(

U (x) = U1 (x) + U2 (x) + Us (x)

(5)

with 1

2

(x) ; Us (x) = (us (x) ; 0) (6) 1 2 s The displacement …elds u (x) ; u (x) and u (x) satisfy the following vectorial Helmholtz equations U1 (x) = u1 (x) ;

(x) ; U2 (x) = u2 (x) ;

+ k12 u1 (x) = 0;

+ ks2 us (x) = 0

+ k22 u2 (x) = 0;

(7)

The temperature …elds satisfy the scalar Helmholtz equations + k12

1

(x) = 0;

+ k22

2

(8)

(x) = 0

The dispersion relations characterizing equation (4) are written as k12 + k22 = q (1 + ) + kp2 ; k12 k22 = q kp2 ;

ks2 =

!2

(9)

where k1 and k2 are the complex wavenumbers of the elastothermal and thermoelastic waves respectively and are given by kj = !=vj + idj ; vj > 0; dj > 0; j = 1; 2 vj are the phase velocities and dj determine the corresponding dissipation cop = is the wavenumber of the uncoupled transverse wave; e¢ cients, k = ! p s kp = ! = ( + 2 ) is the wavenumber of the longitudinal wave in the absence of thermal interactions and = = ( + 2 ) is the dimensionless thermoelastic coupling constant. From the dispersion relations it comes out 4

LOW FREQUENCES IN THERMOELASTICITY

87

that the transverse …eld is not a¤ected by the existence of the thermal …eld. The longitudinal …eld gives birth to two curl free …elds, u1 (x) and u2 (x), which are characterized as the elastothermal and thermoelastic …eld respectively and in view of (9) are involved in the thermoelastic propagation of waves. On the surface of the dislocation we assume that one of the following boundary conditions holds Bj (@x ; b ) U (x) = 0; x 2 ; j = 1; 2; 3; 4

(10)

U (x) = Uin (x) + Usc (x)

(11)

where the uni…ed thermoelastic total …eld U (x) is given by

Uin (x) is the incident …eld, which is an entire solution of (4), and Usc (x) is b the scattered …eld. If the incident …eld is a plane which propagates in the d direction then it admits the following form Uin (x) = A1

b d

ei

1

where

1

=

i k1 q k12 q

;

1

=

bx k1 d ik2

(k12

b 1

2d

+ A2

k22 )

ei

bx k2 d

b b 0

+ A3

ei

bx ks d

(12)

:T hef ourboundarydif f erentialoperatorsBj

are given by B1 (@x ; b ) =

B2 (@x ; b ) = R (@x ; b ) = B3 (@x ; b ) =

B4 (@x ; b ) =

I3 0 0 1

(13)

T (@x ; b ) 0

@v

I3 0 0 @v

T (@x ; b ) 0

b

(14) (15)

1

b

(16)

I3 is 3x3 unit matrix and the operator T (@x ; b ) is the surface traction operator of elasticity and is given by T (@x ; b ) = 2 b r + 5

br +

b

r

(17)

88

GINTIDES-KIRIAKI

The …rst thermoelastic problem corresponds to a rigid screen at constant temperature, it is of Dirichlet type, the second to a screen with a Neumann type condition in thermal insulation, the third to a Dirichlet condition in thermal insulation and the fourth to a Neumann type problem at a constant temperature. For the well posedness of the exterior boundary value problems, the scattered …eld Usc (x) must satisfy the Kupradze radiation conditions [16] as r = jxj ! 1 for i = 1; 2; 3 and j = 1; 2 1 1 ; @xi uj (x) = O 2 ; r r 1 1 j (x) = o ; @xi j (x) = O 2 r r 1 us (x) = O ; r (@xi us (x) i ks us (x)) = O r uj (x) = o

1 r

(18)

Uniqueness and existence theorems for thermoelastic screens in R3 have been proved by Cakoni [7].

3

The equivalent model at low frequencies

In what follows, we will consider the …rst two boundary value problems corresponding to the boundary operators (13) and (14). For the remaining boundary conditions, corresponding to the boundary operators (15) and (16), the consideration is similar and all the remaining analysis is straightforward. Let us assume that the uni…ed thermoelastic vector U (x) and the surface traction - ‡ux vector R (@x ; b ) U (x) be discontinuous across . We de…ne as usually with the bracket notation the discontinuities of the thermal displacement …eld [U (x)] := U+ (x) U (x) (19) and the thermal stress …eld [R (@x ; b ) U (x)] :=

R (@x ; b ) U+ (x)

We consider a su¢ ciently large domain

6

R (@x ; b ) U (x)

(20)

containing the dislocation . Using

LOW FREQUENCES IN THERMOELASTICITY

89

the thermoelastic Green’s integral formula [16] in , we have Z Z (W L U U L W) dx = W [R U] ds n + Z Z [U] R Wds + (W R U U R W) ds

(21)

@

+

where R is the adjoint di¤erential operator to R and can be applied by interchanging with i ! . The uni…ed thermoelastic fundamental solution ~ (x; x0 ) satis…es the equation E ~ (x; x0 ) = LE

4

(x

x0 ) ~I4

(22)

and its explicit form can be found in [7]. Applying Green’s formula (21) for ~ (x; x0 ) and the …eld U and taking into account the fundamental solution E (22) we conclude to Z ~ (x; x0 ) [R U (x0 )] ds (x0 ) U (x) = E Z + ~ T (x; x0 ) ds (x0 ) [U (x0 )] R E Z + ~ (x; x0 ) RU (x0 ) U (x0 ) R E ~ T (x; x0 ) ds (x0 ) E + @

(23)

From the properties of the delta function, equation (23) yields Z Z 0 ~ (x; x ) U (x) = E (x0 y) [R U (y)] ds (y) Z

+

Z

+

[U (y)] (R

(x0

y)) ds (y)

dx0

+

~ (x; x0 ) RU (x0 ) E

U (x0 ) R

~ T (x; x0 ) ds (x0 ) E

@

(24)

From equation (24) we infer that any boundary value problem concerning a dislocation problem on ; can be considered as an equivalent nonhomogeneous problem LU (x) =

[R U (x0 )] 7

[U (x0 )] (R

) ; x 2R3

(25)

90

GINTIDES-KIRIAKI

where is the Dirac measure on : Now, if we have the …rst boundary value problem, the …eld U (x0 ) vanishes on which means that [U (x0 )] = 0 on . So, for this case only the discontinuity of the uni…ed thermal stress …eld remains in (25). Similarly, for the second boundary value problem, only the second term remains in the right hand side of (25) . In what follows we will examine the problem in the frame of low frequencies. We can consider problems in the low frequency regime in cases where the characteristic dimension of the scatterer is much less than the wavelength of the incident …eld [12]. Let us now consider the thermal stress dislocation. In the right hand side of (25) only the term concerning the discontinuity of the stress …eld appears because the other term vanishes due to the continuity of the displacement …eld on . The low frequency properties of the Biot system of thermoelasticity is presented in [12]. In the low frequency limit we have ! ! 0 and this means that all wavenumbers tend to 0. We de…ne the constants p q kp km k = ks ; p = ; q = ; m= ; m = 1; 2 ks ks ks Using this notation, it is shown in [12] that the scattered …eld U (x) can be expanded in the form U (x) =

1 X (i k)n n=0

n!

Un (x)

for certain vector functions Un (x). The incident plane wave of the form (12) assumes the following expansion: 2 0 1 0 n n 1 1 b d b x b b n b b d d d x X 2 (i k) 4 1 n @ 2 n Uin (x) = A 1 n 1 A+A 2 @ n 1 A n! b b b b n d x n d x n=0 1 n ! b d b x b d +As (26) 0

We observe that the …rst term in the expansion is only of elastic character, that is Uin (x) = U0 (x) + O (k) where Uin;0 (x) = A1

0 1

b d 0

+ A2 8

0 2

b 0

2d

+ As

b d 0

LOW FREQUENCES IN THERMOELASTICITY

91

Since we are interested to consider scattering problems in the low frequency region, we assume that the incident …eld is approximately constant. The same result holds also for the stress …eld, since the stress tensor has …rst order derivatives, the …rst order terms are deduced from Uin;1 (x) where the b x b produce only constant terms. From application of the derivatives on d this argument we can assume that the non - homogeneous term in (25) for the thermal stress dislocation is approximately the same at all points on the screen. So, supposing that [R U (x0 )] = B where B =

b

(27)

is a constant vector, equation (25) is written as

b

B ; x 2R3

LU (x) =

(28)

For the case of the thermal displacement dislocation, assuming again that at low frequencies the displacement …eld is constant at all points on the screen, equation (25) yields the form LU (x) = where A =

a

) ; x 2R3

A (R

(29)

is an unknown constant vector.

a

We next derive an identity which is crucial to the reconstruction method. For any C2 vector W using the corresponding Green’s formula [16] and equation (25), we conclude that Z Z 1 0 0 0 W (x ) [R U (x )] ds (x ) + U L Wdx j j n Z = (U (x0 ) R W (x0 ) W (x0 ) R U (x0 )) ds (x0 ) (30) @

where j j is the area of be written as 1

=

j j Z +

Z

Z@ @

w

. Assuming that W :=

equation (30) can

w

(w b + (w (T u (u (T w

b ) ds

w

+

Z

qk b ) + b

9

w

)+

U L Wdx n w

b

b

r ) ds r

w

) ds

(31)

92

GINTIDES-KIRIAKI

4

Uniqueness of the inverse thermoelastic problem for a screen

This uniqueness of the inverse scattering problem is strongly related with the unique continuation principle. It is well known that for elliptic equations the unique continuation principle is equivalent with the uniqueness of the Cauchy problem. In our case, the uniqueness for the Cauchy problem is satis…ed since we have homogeneous and isotropic thermoelastic medium, the coe¢ cients are analytic and Holmgren’s theorem can be applied [17]. The inverse problem we face is the determination of the shape of the planar screen and the vector B if we have a thermal stress dislocation or the vector A if we have displacement dislocation. These vectors characterize the physical properties of the dislocation. We assume that the available information is the knowledge of the …eld U on the surface of a sphere totally enclosing the planar screen. We assume that we know a priori the type of the unknown screen that is we know that we have a thermal stress dislocation or a displacement dislocation. We prove the following uniqueness theorem for this inverse thermal stress dislocation problem: Theorem 1 Let 1 and 2 be two planar screens and B1 ; B2 be two constant vectors. Assume that Uj ; j = 1; 2 are solutions of the scattering problem: LUj (x) =

j

Bj ; j = 1; 2; x 2R3

(32)

and the …elds are satisfying the radiation conditions (18). If U1 = U2 on @ then 1 = 2 and B1 = B2 . Proof. Suppose 1 6= 2 and we have two …elds U1 and U2 such that U1 = U2 on @ . Then there exists a point x0 2 1 such that x0 2 = 1 . It is well known that solutions of the thermoelasticity for a homogeneous and isotropic material admit the unique continuation property [17]. So, from equality of the vector …elds U1 = U2 on @ and unique continuation we infer that there exists a constant 0 such that U1 (x0 + b ) = U2 (x0 + b ) ; 0
0, a > 0, −∞ < r < ∞ and s > 0. The paper discussed various properties of (1) with applications. However, little was presented in terms of basic mathematical properties. The only mathematical properties of substance discussed in Bousquet et al. (2006) were the expected value (E(X)), variance (V ar(X)) and the moment generating function (mgf) (M (t) = E[exp(tX)]) all for the particular case r = 2. Here, I would like to point out that much more general expressions can be derived for the moments and the mgf associated with (1). The main results are presented in Section 2. The calculations involve the generalized hypergeometric function defined by ∞ X (a1 )k (a2 )k · · · (ap )k xk , p Fq (a1 , . . . , ap ; b1 , . . . , bq ; x) = (b1 )k (b2 )k · · · (bq )k k! k=0

where (c)k = c(c+1) · · · (c+k −1) denotes the ascending factorial. The properties of the generalized hypergeometric function can be found in Prudnikov et al. (1986) and Gradshteyn and Ryzhik (2000).

2

Main Results

Theorem 1 derives explicit expressions for E(X γ ) associated with (1), where r and γ can be any real numbers. The corresponding mgf is given by Theorem 2. An essential assumption of the theorems is that the parameter r is a rational number.

1

302

NADARAJAH

Theorem 1 Suppose a random variable X has the cdf (1). Then, the γth moment of X is given by  γI(γ, a, r, s), if r > 0 and γ > 0, γ E (X ) = γJ(γ, a, −r, s), if r < 0 for a > 0 and s > 0, where I(· · ·) and J(· · ·) are given by Lemmas 1 and 2, respectively. Proof: The result follows by writing γ

Z

E (X ) = γ

∞

xγ−1 exp (−axr − sx) dx

0

and using the results of Lemmas 1 and 2.  Theorem 2 Suppose a random variable X has the cdf (1). Then, the mgf of X is given by  arI(r, a, r, s − t) + sI(1, a, r, s − t), if r > 0, M (t) = arJ(r, a, −r, s − t) + sJ(1, a, −r, s − t), if r < 0 for a > 0 and s > t, where I(· · ·) and J(· · ·) are given by Lemmas 1 and 2, respectively. Proof: The probability density function (pdf) corresponding to (1) is:
fX (x) = arxr−1 + s exp (−axr − sx) . Thus, the mgf can be expressed as Z ∞ Z M (t) = ar xr−1 exp {−axr − (s − t)x} dx + s 0

∞

exp {−axr − (s − t)x} dx.

0

The result follows by Lemmas 1 and 2. 

Appendix The proofs of Theorems 1 and 2 require the following technical lemmas. Lemma 1 (Equation (2.3.1.13), Prudnikov et al., 1986, volume 1) For r > 0, γ > 0, a > 0 and s > 0, Z ∞ xγ−1 exp (−axr − sx) dx = I(γ, a, r, s), 0

where  q−1  X  (−a)j   Γ (γ + rj) p+1 Fq (1, ∆ (p, γ + rj) ; ∆(q, 1 + j); (−1)q z) , if 0 < r < 1,  γ+rj   j!s  j=0   p−1       X γ+h (−s)h γ+h (−1)p I = Γ F 1, ∆ q, ; ∆(p, 1 + h); , if r > 1,  q+1 p (γ+h)/r  r r z  rh!a  h=0    Γ(γ)   , if r = 1  (a + s)γ provided that r = p/q, where p ≥ 1 and q ≥ 1 are co-prime integers, where z = pp aq /{sp q q } and ∆(k, a) = (a/k, (a + 1)/k, . . . , (a + k − 1)/k). 2

B-DISTRIBUTION

303

Lemma 2 (Equation (2.3.1.14), Prudnikov et al., 1986, volume 1) For r > 0, a > 0 and s > 0, Z ∞
xγ−1 exp −ax−r − sx dx = J(γ, a, r, s), 0

where J

=

q−1 X (−a)j Γ (γ − rj) 1 Fp+q (1; ∆ (p, 1 − γ + rj) , ∆(q, 1 + j); z) γ−rj j!s j=0       p−1 X γ+h (−s)h a(γ+h)/r γ+h 1; ∆ q, 1 + , ∆(p, 1 + h); z + Γ − F 1 p+q rh! r r h=0

provided that r = p/q, where p ≥ 1 and q ≥ 1 are co-prime integers, where z = (−1)p+q sp aq /{pp q q } and ∆(k, a) = (a/k, (a + 1)/k, . . . , (a + k − 1)/k).

References Bousquet, N., Bertholon, H. and Celeux, G. (2006). An alternative competing risk model to the Weibull distribution for modelling aging in lifetime data analysis. Lifetime Data Analysis, 12, 481–504. Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products, sixth edition. Academic Press, San Diego. Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series, volumes 1, 2 and 3. Gordon and Breach Science Publishers, Amsterdam.

3

304

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.3,305-318,2008, COPYRIGHT 2008 EUDOXUS PRESS, 305 LLC

The Beta-Laplace Distribution Tomasz J. Kozubowski, Department of Mathematics & Statistics, University of Nevada, Reno, NV 89557, USA, [email protected]

Saralees Nadarajah, Department of Statistics, University of Nebraska, Lincoln, NE 68583, USA, [email protected] Abstract. Motivated by the recent work of Eugene et al. [3] and Gupta and Nadarajah [5], we introduce the beta Laplace distribution generated from the logit of a beta random variable. Our focus are the basic theoretical properties of this distribution, including modality and concavity of the density, moments and related parameters, and stochastic representations that aid in random variate generation from this model. Keywords: Beta-normal distribution, Gauss hypergeometric exponential distribution, Generalized exponential distribution, Mixture representation, Unimodality AMS 2000 Subject Classifications: 60E05, 62E10

1

Introduction

Every cumulative distribution function (c.d.f.) G generates a generalized class of distributions with c.d.f.’s FG (x) =

BG(x) (a, b) , x ∈ R, a, b > 0, B(a, b)

where By (a, b) =

Z y

wa−1 (1 − w)b−1 dw

0

is the incomplete beta function and B(a, b) = B1 (a, b) =

Γ(a)Γ(b) . Γ(a + b)

(1)

(2)

306

KOZUBOWSKI-NADARAJAH

Note that if a random variable X has the c.d.f. given by (1) then it admits the representation d

X = G−1 (W ),

(3)

where W has the beta distribution with parameters a > 0, b > 0, denoted by Be(a, b). In the special case a = b = 1 the variable W is standard uniform, and (3) becomes the probability integral transformation. Eugene et al. [3] introduced what is known as the beta normal distribution by taking G to be the c.d.f. of the normal distribution with parameters µ and σ. The properties of this distribution have been studied in more detail in [5]. In this paper, we introduce the beta Laplace (BL) distribution by taking G in (1) to be the c.d.f. of the Laplace distribution. Thus, the c.d.f. of the BL distribution is given by (1), where

G(x)

   1 x−θ   , if x < θ,  exp 2 σ  = θ−x 1    1 − exp , if x ≥ θ,

2

(4)

σ

and −∞ < θ < ∞ and σ > 0. The corresponding probability density function (p.d.f.) and the hazard rate function are 



1 |x−θ | exp − Ga−1 (x) {1 − G(x)}b−1 2σB(a, b) σ

fa,b,θ,σ (x) =

(5)

and λa,b,θ,σ (x) =



1 |x−θ | exp − 2σ[B(a, b)]2 φ



Ga−1 (x) {1 − G(x)}b−1 , B1−G(x) (b, a)

(6)

respectively. We shall denote this distribution by BL(a, b, θ, σ). Since θ and σ are location and scale parameters, we shall restrict attention to the standard case with θ = 0 and σ = 1, denoted by BL(a, b). In this case the p.d.f. (5) takes the form fa,b (x) =

 a+b−1

1 2

   eax (2 − ex )b−1 ,

if x ≤ 0, Γ(a + b) ·  −bx Γ(a)Γ(b)  e (2 − e−x )a−1 , if x > 0.

(7)

Note that X ∼ BL(a, b, θ, σ) if and only if −X ∼ BL(b, a, −θ, σ). In particular, for the standard variable we have X ∼ BL(a, b) if and only if −X ∼ BL(b, a).

BETA-LAPLACE DISTRIBUTION

307

The standard Laplace distribution is contained as the particular case of (1) for a = 1 and b = 1. Another special case is FG (x)

! n X n = Gi (x) {1 − G(x)}n−i i=a

i

for b = n − a + 1 and integer values of a. This is the distribution of the ath order statistic connected with a random sample of size n = a + b − 1 from the Laplace distribution with c.d.f. G, see [2]. Basic properties of this distribution can be found in Section 2.5 of [7]. Other special cases include FG (x) = 1 −

a {1 − G(x)}b X Γ(b + i − 1) i−1 G (x) Γ(b) Γ(i) i=1

for integer values of a, FG (x) =

b Ga (x) X Γ(a + i − 1) {1 − G(x)}i−1 Γ(a) i=1 Γ(i)

for integer values of b, and FG (x) =

2 arctan π

s

G(x) 1 − G(x)

for a = 1/2 and b = 1/2. In this paper we derive basic theoretical properties of (5), deferring practical issues of estimation and testing to future work. In particular, in Section 2 we derive stochastic representations of the corresponding random variables, which aid in simulation from this distribution, characterize the corresponding densities in terms of modality and concavity in Section 3, and provide expressions for the moments and related characteristics in Section 4. Proofs and technical lemmas are collected in Section 5.

2

Representations

Note that if X ∼ BL(a, b), then the distribution of X|X > 0 (the distribution of X truncated below at zero) is given by the p.d.f. ga,b (x) =

(1/2)a+b−1 −bx e (2 − e−x )a−1 , x ≥ 0, a, b > 0, B1/2 (b, a)

(8)

308

KOZUBOWSKI-NADARAJAH

where B1/2 (b, a) is the incomplete beta function given by (2). This “one-sided” beta-Laplace distribution is a generalization of the standard exponential distribution, similar in spirit to the generalized exponential distribution introduced in [4]. The latter can be defined through the representation (3), where W has the power function distribution with the c.d.f. wα , 0 < w ≤ 1, α > 0, while G(x) = 1 − e−x , 0 < x ≤ 1, is the c.d.f. of the standard exponential distribution (so it reduces to the exponential when α = 1). Similarly, a variable with p.d.f. (8) admits such a representation with the same G and W having the p.d.f. wa,b (x) =

(1/2)a+b−1 (1 − x)b−1 (1 + x)a−1 , 0 ≤ x ≤ 1, a, b > 0. B1/2 (b, a)

(9)

The distribution with density (9) is a special case of Gauss hypergeometric distribution studied in [1] (see also [6], p. 253). For this reason we shall refer to the distribution with density (8) as the Gauss hypergeometric exponential distribution with parameters a, b > 0, denoted by GHE(a, b). Note that when a = b = 1 this distribution reduces to the standard exponential as in this case W is standard uniform. Similarly, when we truncate X ∼ BL(a, b) above at zero, we obtain a distribution on (−∞, 0) corresponding to a r.v. −Y , where Y has a GHE(b, a) distribution on (0, ∞). This leads to the following representation of X in terms of its “one-sided” counterparts. Proposition 1 If X ∼ BL(a, b) then we have d

X = IY1 + (I − 1)Y2 ,

(10)

where Y1 ∼ GHE(a, b), Y2 ∼ GHE(b, a), I takes on the values 0 and 1 with probabilities qa,b (x) =

Γ(a + b) Γ(a + b) B1/2 (a, b) and pa,b = 1 − qa,b (x) = B (b, a), Γ(a)Γ(b) Γ(a)Γ(b) 1/2

(11)

respectively, and all the variables on the right-hand-side of (10) are mutually independent. As we stated before, the r.v.’s Y1 and Y2 in the above representation have the same distributions as F −1 (W1 ) and F −1 (W2 ), respectively, where F is the standard exponential c.d.f. and W1 and W2 have densities given by wa,b and wb,a , respectively. Further, W1 and W2 have the

BETA-LAPLACE DISTRIBUTION

309

same distributions as 2V1 − 1 and 1 − 2V2 , respectively, where V1 and V2 have truncated beta distributions, that is d

d

V1 = W |W > 1/2, and V2 = W |W < 1/2

(12)

with W ∼ Be(a, b). Moreover, the quantities qa,b and pa,b in (11) are the probabilities P (W < 1/2) and P (W > 1/2), respectively. This result admits a generalization, where X is still given by (3) but G and W are not necessarily the Laplace c.d.f. and the beta variable, respectively. Indeed, let V have a continuous distribution on (0, ∞) with density f and c.d.f. F , and let Y be the corresponding “symmetrization” with p.d.f. g(x) = f (|x|), x ∈ R, and c.d.f. G(x) =

 1   (1 − F (−x)) ,

if x < 0, 2 1   1 − (1 − F (x)) , if x ≥ 0. 2

(13)

[If V is standard exponential then Y is standard Laplace (4) with θ = 0 and σ = 1.] Further, define a r.v. X via (3) where W has some continuous distribution on (0, 1). Then the following representation holds. Proposition 2 The r.v. X ∼ G−1 (W ) admits the representation (10), where d

d

Y1 = F −1 (2V1 − 1) and Y2 = F −1 (1 − 2V2 )

(14)

with V1 , V2 given by (12), I takes on the values 0 and 1 with probabilities P (W < 1/2) and P (W > 1/2), respectively, and all the variables on the right-hand-side of (10) are mutually independent.

3

Characterization of the density

Before we describe the density (7) of the standard beta-Laplace distribution in terms of modality and concavity, we start with one-sided distributions given by (8). Thus, we consider the function ha,b (x) = e−bx (2 − e−x )a−1 , x ≥ 0, a, b > 0,

(15)

310

KOZUBOWSKI-NADARAJAH

which appears in the expression for the density. The relevant properties of ha,b are as follows. Proposition 3 The function ha,b given by (15) admits the following properties: (i) The function ha,b is continuous on [0, ∞) and differentiable on (0, ∞) with ha,b (0) = 1, limx→∞ ha,b (x) = 0, and limx→0+ h0a,b (x) = a − b − 1. (ii) The function ha,b is monotonically decreasing on (0, ∞) whenever a > 0 and b ≥ a − 1, and is monotonically increasing on (0, xa,b ) and decreasing on (xa,b , ∞) whenever a > 1 and b < a − 1, where a+b−1 , 0 < b < a − 1. 2b

xa,b = ln

(16)

Moreover, in the latter case the maximum value of ha,b is ha,b (xa,b ) = 2a+b−1

(a − 1)a−1 bb , 0 < b < a − 1. (a + b − 1)a+b−1

(17)

(iii) If either a > 3 and b− (a) ≤ b < b+ (a) or 1 < a ≤ 3 and b < b+ (a) then ha,b is concave + down on (0, x+ a,b ) and concave up on (xa,b , ∞). Here,

b± (a) = a − 1 ± and x± a,b where t± a,b

=

(a − 1)(1 + 2b) ±

= ln

q

2(a − 1)

1 + t± a,b 2

,

p

(a − 1)2 (1 + 2b)2 − 4b2 (a − 1)(a − 2) . 2b2

(18)

(19)

(20)

+ (iv) If a > 3 and b < b− (a) then ha,b is concave down on (x− a,b , xa,b ) and concave up on + (0, x− a,b ) and (xa,b , ∞).

(v) If either 1 < a and b ≥ b+ (a) or a ≤ 1 then ha,b is concave up on (0, ∞). Using the results above we obtain the following four distinct cases of the one-sided BL density ga,b given by (8): • If either 0 < a ≤ 1 or a > 1 but b ≥ b+ (a) then ga,b (x) is concave up and decreasing in x on (0, ∞).

BETA-LAPLACE DISTRIBUTION

311

• If a > 1 and a − 1 ≤ b < b+ (a) [note that b+ (a) > a − 1 whenever a > 1] then ga,b (x) is decreasing in x on (0, ∞). Moreover, it is concave down on (0, x+ a,b ) and concave up + on (x+ a,b , ∞) with xa,b as in (19).

• If either 1 < a ≤ 3 and 0 < b < a − 1 or a > 3 and b− (a) ≤ b < a − 1 [note that b− (a) < a − 1 whenever a > 3] then ga,b (x) is increasing in x on (0, xa,b ), decreasing in + x on (xa,b , ∞), concave down on (0, x+ a,b ), and concave up on (xa,b , ∞), with 0 < xa,b
0.

we obtain the following result. Proposition 4 For any a, b > 0 the BL(a, b) distribution is unimodal and we have the following three distinct cases: (i) If a − 1 ≤ b ≤ a + 1 then the density (21) is monotonically increasing on (−∞, 0) and decreasing on (0, ∞) (so that the mode occurs at x = 0). (ii) If 0 < b < a − 1 then the density (21) is monotonically increasing on (−∞, xa,b ) and decreasing on (xa,b , ∞) (so that the mode occurs at x = xa,b > 0 given by (16)). (iii) If 0 < a < b − 1 then the density (21) is monotonically increasing on (−∞, −xb,a ) and decreasing on (−xb,a , ∞) (so that the mode occurs at x = −xb,a < 0), where xb,a = ln

a+b−1 , 0 < a < b − 1. 2a

(22)

312

KOZUBOWSKI-NADARAJAH

Moreover, in cases (ii) and (iii) the maximum values of fa,b are fa,b (xa,b ) =

Γ(a + b) (a − 1)a−1 bb Γ(a + b) (b − 1)b−1 aa , and f (−x ) = , (23) a,b b,a Γ(a)Γ(b) (a + b − 1)a+b−1 Γ(a)Γ(b) (a + b − 1)a+b−1

respectively. Remark 1. Note that by Part (i) of Proposition 3 the BL density is continuous on R but differentiable only at x 6= 0 (regardless of whether the mode is at zero or not). Remark 2. Note that, in contrast, the beta normal distribution introduced in [3] is not always unimodal. Remark 3. Combining Propositions 3 and 4 and using the representation (21) one can derive a number of distinct cases of the density fa,b in terms of its modality and concavity.

4

Moments and related measures

We start with the moment generating function (m.g.f.) corresponding to the BL(a, b) distribution. By (7) we have  Z ∞ e−(a+t)x tX M (t) = E(e ) = C  0 2a

e−x 1− 2

!b−1

dx +

Z ∞ −(b−t)x e 0

2b

e−x 1− 2

!a−1

 

dx , 

(24)

where C = Γ(a + b)/[Γ(a)Γ(b)]. Both integrals above are convergent whenever −a < t < b. Since e−x /2 ∈ (0, 1/2) when x ∈ (0, ∞), using the binomial expansion (1 + z)p =

∞ X

pjz j , |z| < 1, p ∈ R,

(25)

j=0

we can write (24) as   Z ∞X ∞ ∞ Z ∞ X  j j (−1) (−1) −(a+t+j)x −(b−t+j)x b − 1je dx + a − 1je dx . C  0  2a+j 2b+j 0 j=0

j=0

(26)

BETA-LAPLACE DISTRIBUTION

313

Since the quantities under the summations are absolutely integrable (when −a < t < b), interchanging the order of integration and summation leads to   ∞ ∞ X  X 1 1 (−1)j (−1)j M (t) = C + b − 1j a − 1j , −a < t < b.  2a+j a+t+j 2b+j b−t+j j=0

(27)

j=0

If a = k + 1 and b = m + 1, where k and m are nonnegative integers, then b − 1j = 0 for j > m and a − 1j = 0 for j > k

(28)

so that the series above have only finite number of terms and we have   m k X  X 1 1 (−1)j (−1)j + mj kj M (t) = C  2k+1+j k + 1 + t + j j=0 2m+1+j m + 1 − t + j  j=0

(29)

for −k − 1 < t < m + 1. Note that when k = m = 0 (so that a = b = 1) the above expression simplifies to (1 − t2 )−1 , which is the m.g.f. corresponding to the standard Laplace distribution. Similar derivations lead to the characteristic function (ch.f.), which is of the form   ∞ ∞ X  j j X 1 1 (−1) (−1) + b − 1j a − 1j , t ∈ R. (30) φ(t) = EeitX = C  2a+j a + it + j 2b+j b − it + j  j=0

j=0

Differentiating either the m.g.f. or the ch.f. n times and evaluating at t = 0 we obtain the following expression for the nth moment µn = EX n of X ∼ BL(a, b), where n = 1, 2, . . ., 



∞ ∞  X Γ(a + b) X 1 1 (−1)j+n (−1)j n! b − 1j + a − 1j . µn = Γ(a)Γ(b) j=0 2a+j (a + j)n+1 j=0 2b+j (b + j)n+1 

(31)

As before, if a = k + 1 and b = m + 1, where k and m are nonnegative integers, the series above have only finite number of terms and we have µn =

 r

where r = k + m + 1.

1 2





m k (−1)n+m−j 2j X (−1)k−j 2j  r!n! X mj + kj , k!m! j=0 (r − j)n+1 (r − j)n+1  j=0

(32)

314

KOZUBOWSKI-NADARAJAH

5

Proofs

Proof of Proposition 1. This is a special case of Proposition 2. Proof of Proposition 2. We proceed by showing that for each x ∈ R the c.d.f. of G−1 (W ) coincides with the c.d.f. of the right-hand-side of (10) with Y1 , Y2 , and I as stated above. Assume first that x > 0. Since G−1 (p) =

   −F −1 (1 − 2p) , if p ∈ (0, 1/2),   F −1 (2p − 1) ,

(33)

if p ∈ [1/2, 1),

and G(0) = 1/2, we have P (G−1 (W ) > x) = P (W > 1/2 and F −1 (2W − 1) > x) = P (W > 1/2 and 2W − 1 > F (x)) = P (W > 1/2 and W > (1 + F (x))/2) = P (W > (1 + F (x))/2) as W > (1 + F (x))/2 implies W > 1/2. On the other hand, we have P (IY1 + (I − 1)Y2 > x) = P (IF −1 (2V1 − 1) > x and I = 1) = P (2V1 − 1 > F (x))P (I = 1) = P (V1 > (1 + F (x))/2)P (I = 1) = P (W > (1 + F (x))/2|W > 1/2)P (W > 1/2), which also simplifies to P (W > (1 + F (x))/2). We thus established the equality of the c.d.f.’s when x > 0. The case x ≤ 0 is analogous. This concludes the proof. The following lemma is needed to prove Proposition 3. Lemma 1 Consider the function u(t) = b2 t2 − (a − 1)(1 + 2b)t + (a − 1)(a − 2), t ∈ R,

(34)

where a > 1 and b > 0. Then (i) The function u is decreasing on (−∞, vab ) and increasing on (va,b , ∞), where va,b =

(a − 1)(1 + 2b) > 0. 2b2

(35)

BETA-LAPLACE DISTRIBUTION

315

Moreover, the minimum value of u on (−∞, ∞) is equal to (

u(va,b ) = −(a − 1) (a − 1)

"

1 + 2b 2b

2

#

)

−1 +1

< 0.

(36)

(ii) We have u(0) = (a − 1)(a − 2), so that u(0) < 0 when a < 2, u(0) = 0 when a = 2, and u(0) > 0 when a > 2. (iii) The equation u(t) = 0 admits the roots t± a,b given by (20). (iv) For any fixed a > 1, as a function of b the larger root, t+ a,b , is continuous and decreasing + + + on (0, ∞) with limb→0+ t+ a,b = ∞ and limb→∞ ta,b = 0. Moreover, ta,b = 1 when b = b (a)

with b+ (a) given by (18). (v) For any fixed a > 2, as a function of b the smaller root, t− a,b , is continuous and decreasing − − − on (0, ∞) with limb→0+ t− a,b = ta,0 = a − 2 and limb→∞ ta,b = 0. Moreover, ta,b = 1 when

b = b− (a) with b− (a) given by (18). Proof. Calculations needed to prove Parts (i) - (iii) are elementary. To establish Part (iv), write t+ a,b



1 1 = (a − 1) + 2 2b b



1 + 2b

s

(a −

1)2



2

1 +2 b

− 4(a − 1)(a − 2)

(37)

to see that it is decreasing in b on (0, ∞) and the limits are as stated. With some tedious albeit routine algebra one can check that the solution of the equation t+ a,b = 1 is given by b+ (a) in (18). Similarly, straightforward algebra leads to t− a,b =

2(a − 1)(a − 2) p , (a − 1)(1 + 2b) + (a − 1) {(a − 1)(1 + 4b) + 4b2 }

(38)

which shows that this is a decreasing function of b on (0, ∞) with limits as stated. Further − routine calculations show that the solution of the equation t− a,b = 1 is given by b (a) in (18).

This concludes Part (v). Proof of Proposition 3. The continuity and differentiability of ha,b , as well as the values at zero and infinity, are clear. To check the limit of the derivative, write ha,b (x) = exp(sa,b (x)), where sa,b (x) = ln ha,b (x) = −bx + (a − 1) ln(2 − e−x ), x ∈ [0, ∞),

(39)

316

KOZUBOWSKI-NADARAJAH

so that for x > 0 we have h0a,b (x) = ha,b (x)s0a,b (x) with s0a,b (x) = −b +

a−1 , x ∈ (0, ∞). 2ex − 1

(40)

Since sa,b (x) and s0a,b (x) converge to 0 and a − b − 1, respectively, as x → 0+ , we conclude that the corresponding limit of h0a,b (x) is a − b − 1. This concludes Part (i). To study the monotonicity of the function ha,b we look at the derivate (40) since the signs of s0a,b (x) and h0a,b (x) coincide. It is clear that s0a,b (x) < 0 for all x > 0 whenever α ≤ 1, in which case ha,b is monotonically decreasing on (0, ∞). Further, if a > 1, simple algebra shows that the derivative is negative if and only if ex >

a+b−1 . 2b

(41)

This inequality is always true whenever b ≥ a − 1, while for b < a − 1 (and a > 1) its solution is x > xa,b , where xa,b given by (16). Moreover, routine calculations show that in the latter case the maximum value of ha,b is given by (17). This concludes Part (ii). Next, we study the concavity of ha,b . Observe that h00a,b (x) = ha,b (x)(s00a,b (x) + [s0a,b (x)]2 ), x ∈ (0, ∞),

(42)

so that h00a,b (x) > 0 if and only if s00a,b (x) + [s0a,b (x)]2 > 0.

(43)

Since s00a,b (x) =

−2(a − 1)ex , x ∈ (0, ∞), (2ex − 1)2

(44)

the inequality (43) holds for all x ∈ (0, ∞) whenever a ≤ 1, in which case the function ha,b is concave up on this interval. We thus established the second part of (v). In the sequel, we shall study the inequality (43) assuming that a > 1. Utilizing the expressions for the first and the second derivatives of sa,b , given by (40) and (44), respectively, after some routine algebra we find that this inequality is equivalent to u(2ex − 1) > 0,

(45)

BETA-LAPLACE DISTRIBUTION

where u(·) is the quadratic function defined in Lemma 1. Since for any x > 0 we have 2ex − 1 > 1, we need to check how the roots of the equation u(t) = 0, which by Part (iii) of Lemma 1 are given by (20), relate to t = 1. Observe that by Part (iv) of Lemma 1 the larger + + root (denoted by t+ a,b in (20)) is less than or equal to 1 whenever b ≥ b (a), where b (a) is

given by (18). Consequently, for these values of b we have u(t) > 0 for all t > 1, so that the inequality (45) is satisfied by all x > 0. This shows that whenever a > 1 and b ≥ b+ (a) the function ha,b is concave up on the interval (0, ∞), which is the first part of (v). To establish Parts (iii) and (iv) we shall proceed by separately considering the cases 1 < a ≤ 3 and a > 3. Case 1: 1 < a ≤ 3. If a ≤ 2, then by Parts (i) and (ii) of Lemma 1, we conclude that the smaller root of the equation u(t) = 0, denoted by t− a,b in (20), is less than or equal to zero. Thus, when b < b+ (a), in which case we have t+ a,b > 1 by Part (iv) of Lemma 1, the function u + is negative for t ∈ (0, t+ a,b ) and positive for t ∈ (ta,b , ∞). In turn, the inequality (45) holds for + all x > x+ a,b with xa,b defined in (19). The same conclusion is reached when 2 < a ≤ 3, since

in this case a − 2 ≤ 1 so that by Part (v) of Lemma 1 we have t− a,b ≤ 1. We thus obtained the second part of (iii). Case 2: 3 < a. We shall restrict attention to the case b < b+ (a) (so that the larger root of the equation u(t) = 0 is greater than one), as otherwise we get the case t+ a,b ≤ 1 already considered (where the function ha,b is concave up on (0, ∞)). By Part (v) of Lemma 1 we − − have t− a,b > 1 if and only if b < b (a) with b (a) given by (18). Consequently, with these + x values of b the equation (45) holds whenever 1 < 2ex − 1 < t− a,b (a) or ta,b (a) < 2e − 1 < ∞. + Equivalently, the function ha,b is concave up on the intervals (0, x− a,b ) and (xa,b , ∞), and is + ± concave down on the interval (x− a,b , xa,b ), where xa,b are given by (19). This is Part (iv) of our

result. In turn, when b− (a) ≤ b < b+ (a) then t− a,b ≤ 1 and we similarly conclude that now the function ha,b is concave up on the interval (x+ a,b , ∞) and concave down on the interval (0, x+ a,b ), which the first part of (iii). This concludes the proof.

317

318

KOZUBOWSKI-NADARAJAH

References [1] C. Armero and M.J. Bayarri, Prior assessments for prediction in queues, The Statistician, 43, 139-153 (1994). [2] H.A. David, Order Statistics, John Wiley & Sons, New York, 1981. [3] N. Eugene, C. Lee and F. Famoye, Beta-normal distribution and its applications, Comm. Statist. Theory Methods, 31, 497–512 (2002). [4] R.D. Gupta and D. Kundu, Generalized exponential distributions, Austral. N. Zealand J. Statist., 41(2), 173-188 (1999). [5] A.K. Gupta and S. Nadarajah, On the moments of the beta normal distribution, Comm. Statist. Theory Methods, 33, 1–13 (2004). [6] N.L. Johnson, S. Kotz and B. Balakrishnan, Continuous Univariate Distributions, Vol 2, Second Edition, Wiley, New York, 1995. [7] S. Kotz, T.J. Kozubowski and K. Podg´orski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance, Birkh¨auser, Boston, 2001.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.3,319-331,2008, COPYRIGHT 2008 EUDOXUS PRESS, 319 LLC

On the Stability and Asymptotic Behavior of Generalized Quadratic Mappings ∗ Hark-Mahn Kim Department of Mathematics Chungnam National University 220 Yuseong-Gu, Daejeon, 305-764, Korea [email protected]

ABSTRACT Let E1 and E2 be linear spaces. In this paper we extend a classical quadratic functional equation to more general equations of two types. In addition we solve the generalized Hyers-Ulam-Rassias stability problem for the functional equations, and thus obtain asymptotic properties of quadratic mappings as an application. Keywords: stability, functional equation, generalized quadratic mappings. 2000 Mathematics Subject Classification: 39B82, 39B72.

1 Introduction In 1960 and in 1964 S.M. Ulam [24] proposed the general Ulam stability problem: “When is it true that by slightly changing the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus one can ask the following question for general functional equations: If we replace a given functional equation by a functional inequality, when can one assert that the solutions of the inequality must be close to the solutions of the given equation? If the answer is affirmative, we would say that a given functional equation is stable. In 1978 P.M. Gruber [8] remarked that Ulam’s problem is of particular interest in probability theory and in the case of functional equations of different types. We wish to note that stability properties of different functional equations can have applications to unrelated fields. For instance, Zhou [25] used a stability property of the functional equation f (x − y) + f (x + y) = 2f (x) to prove a conjecture of Z. Ditzian about the relationship between the smoothness of a mapping and the degree of its approximation by the associated Bernstein polynomials. The Ulam’s problem for ε-additive mappings f : E1 → E2 between Banach spaces i.e., kf (x + y) − f (x) − f (y)k ≤ ε for all x, y ∈ E1 , was solved by D.H. Hyers [9] and then generalized by ˇ D.G. Bourgin [4], Th.M. Rassias [17] and P. Gavruta [7] who permitted the Cauchy difference to become unbounded. Now, a square norm on an inner product space satisfies the important parallelogram equality kx + yk2 + kx − yk2 = 2(kxk2 + kyk2 ) for all vectors x, y. If 4ABC is a triangle in a finite

320

KIM

dimensional Euclidean space and I is the center of the side BC, then the following identity −−→ −→ − → −→ kABk2 + kACk2 = 2(kAIk2 + kCIk2 ) holds for all vectors A, B and C. The following functional equation which was motivated by these equations Q(x + y) + Q(x − y) = 2Q(x) + 2Q(y)

(1.1)

is called a quadratic functional equation, and every solution of the equation (1.1) is said to be a quadratic mapping. The quadratic functional equation and several other functional equations are useful to characterize inner product spaces [1, 2, 18, 22]. A Hyers-Ulam stability theorem for the quadratic functional equation was proved by a lot of authors [5, 19]. C. Borelli and G.L. Forti [3] generalized the stability result as follows: Let G be an abelian group, and E a Banach space. Assume that a mapping f : G → E satisfies the functional inequality kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤ ϕ(x, y) for all x, y ∈ G, and ϕ : G × G → [0, ∞) is a function such that ∞ X 1 Φ(x, y) := ϕ(2i x, 2i y) < ∞ i+1 4 i=0

for all x, y ∈ G. Then there exists a unique quadratic mapping Q : G → E satisfying kf (x) −

f (0) − Q(x)k ≤ Φ(x, x) 3

for all x ∈ G. In 1983 F. Skof [23] was the first author to solve the Ulam problem for additive mappings on a restricted domain. In 1998 S. Jung [12] and in 2004 J.M. Rassias [16] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem [10, 11, 14, 15, 20, 21]. Now we are going to extend the equation (1.1) to more generalized equations with (d + 1)U variables. For this purpose, we employ the operator x2 f (x1 ), which is defined in [13] as follows ]

f (x1 ) = f (x1 + x2 ) + f (x1 − x2 )

x2

for a given mapping f : E1 → E2 between vector spaces. Similarly, we define ´ U ³U f (x ) and inductively 1 x2 x3 d ]

] ³

f (x1 ) =

x2 ,··· ,xd+1

xd+1

d−1 ]

U2

x2 ,x3

´ f (x1 )

x2 ,··· ,xd

for all natural number d. Then it is easy to see that the operation

Ud

x2 ,··· ,xd+1

f (x1 ) can be

expressed in the form d ] x2 ,··· ,xd+1

f (x1 ) =

d X k=0

 

 X

2≤i1 0 such that either ( sup

kDf (x1 , x2 , · · · , xd+1 )k :

x1 ,··· ,xd+1

d+1 X

) kxi k ≥ r

i=1

or ( sup

kEf (x1 , x2 , · · · , xd+1 )k :

x1 ,··· ,xd+1

d+1 X

) kxi k ≥ r

i=1

is bounded for all d ≥ 1. Proof. Let supx1 ,··· ,xd+1 kDf (x1 , x2 , · · · , xd+1 )k ≤ M < ∞ for all d ≥ 1. Then for each d ≥ 1, there exists a unique quadratic mapping Qd : X → Y which satisfies the equation (1.3) and the inequality kf (x) − Qd (x)k ≤

3M + 1)

2(2d−1

for all x ∈ X by Theorem 3.1. Let m be a positive integer with m > d. Then, we obtain kf (x) − Qm (x)k ≤

3M 3M ≤ 2(2m−1 + 1) 2(2d−1 + 1)

for all x ∈ X. The uniqueness of Qd implies that Qm = Qd for all m with m > d, and so kf (x) − Qd (x)k ≤

M 3(2m−1 + 1)

for all x ∈ X. By letting m → ∞, we conclude that f is itself quadratic. The reverse assertion is trivial.

¤

Acknowledgment ∗

This work was supported by the Brain Korea 21 Project in 2006.

References [1] J. Acz´ el and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, 1989. ¨ [2] Dan Amir, Characterizations of inner product spaces, Birkhauser-Verlag, Basel, 1986.

330

KIM

[3] C. Borelli and G.L. Forti, On a general Hyers-Ulam stability result, Internat. J. Math. Math. Sci. 18(1995), 229-236. [4] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57(1951), 223-237. [5] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62(1992), 59-64. [6] S. Czerwik, The stability of the quadratic functional equation, in ‘Stability of Mappings of Hyers-Ulam Type’ (edited by Th. M. Rassias and J. Tabor), Hadronic Press, Florida, 1994, pp 81-91. ˇ [7] P. Gavruta, A generalization of the Hyers-Ulam-Rassias Stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [8] P.M. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245(1978), 263-277. [9] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27(1941), 222-224. [10] K. Jun and H. Kim, On the Hyers-Ulam stability of a difference equation, J. Comput. Anal. Appl. 7(4)(2005), 397-407. [11] K. Jun, H. Kim and I. Chang, On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, J. Comput. Anal. Appl. 7(1)(2005), 21-33. [12] S. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222(1998), 126-137. [13] H. Kim, On the stability problem for a mixed type of quartic and quadratic functional equation, J. Math. Anal. Appl. 324(2006), 358-372. [14] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46(1982), 126-130. [15] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory, 57(1989), 268-273. [16] J.M. Rassias, Asymptotic behavior of mixed type functional equations, Austral. J. Math. Anal. Appl. 1(1)(2004), 1-21. [17] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297-300. [18] Th. M. Rassias, Inner product spaces and applications, Longman, 1997. [19] Th.M. Rassias, On the stability of the quadratic functional equation and its applicattions, Studia, Univ. Babes-Bolyai, XLIII (3) (1998), 89-124. [20] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251(2000), 264-284.

GENERALIZED QUADRATIC MAPPINGS

331

[21] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62(2000), 23-130. [22] D.A. Senechalle, A characterization of inner product spaces,

Proc. Amer. Math. Soc.

19(1968), 1306-1312. [23] F. Skof, Sull’ approssimazione delle applicazioni localmente δ-additive, Atti Accad. Sci. Torino Cl Sci. Fis. Mat. Natur. 117(1983), 377-389. [24] S.M. Ulam, A collection of the mathematical problems, Interscience Publ. New York, 1960. [25] Ding-Xuan Zhou, On a conjecture of Z. Ditzian, J. Approx. Theory 69(1992), 167-172.

332

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.3,333-344,2008, COPYRIGHT 2008 EUDOXUS PRESS, 333 LLC

ON STATISTICAL FUZZY TRIGONOMETRIC KOROVKIN THEORY OKTAY DUMAN AND GEORGE A. ANASTASSIOU

Abstract. In this study, we use regular matrix transformations in the approximation by fuzzy positive linear operators, where the test functions are trigonometric. So we prove a trigonometric fuzzy Korovkin theorem by means of A-statistical convergence, where A is a non-negative regular summability matrix. We also study rates of A-statistical convergence of a sequence of fuzzy positive linear operators in the trigonometric environment.

1. Introduction The study of the Korovkin type approximation theory is an area of active research, which deals with the problem of approximating a function by means of a sequence of positive linear operators. In recent years, this theory has been improved with the help of two di¤erent ways. The …rst one, statistical approximation, is to use the notion of statistical convergence in the approximation by operators. Earlier studies show that the statistical approximation enables us to obtain more powerful results than the classical aspects (see, for instance, [1, 2, 3, 4, 5]). The second one is to obtain fuzzy approximation by fuzzy positive linear operators via the concept of fuzzy set theory (see, [6, 7, 8, 9, 10]). Our primary interest of the present paper is to combine these ways: statistical approximation and fuzzy approximation. So, we obtain a statistical fuzzy Korovkin-type approximation theorem and compute its statistical rates in the approximation. Here the test functions are trigonometric. We should note that, as a rule, neither limits nor statistical limits can be calculated or measured with absolute precision. To re‡ect this imprecision several approaches in mathematics have been developed: fuzzy set theory, fuzzy logic, interval analysis, set valued analysis, etc. We …rst collect some basic de…nitions and results used in the paper. A fuzzy number is a function : R ! [0; 1]; which is normal, convex, upper semi-continuous and the closure of the set supp( ) is compact, where supp( ) := fx 2 R : (x) > 0g: The set of all fuzzy numbers are denoted by RF . Let [ ]0 := fx 2 R : (x) > 0g and [ ]r := fx 2 R : (x)

rg; (0 < r

1):

Then, it is well-known [11] that, for each r 2 [0; 1]; the set [ ]r is a closed and bounded interval of R. For any u; v 2 RF and 2 R, it is possible to de…ne uniquely the sum u v and the product u v as follows: [u

v]r = [u]r + [v]r and [

v]r = [u]r ; (0

r

1):

Key words and phrases. Statistical convergence, statistical rates, fuzzy positive linear operators, trigonometric functions, trigonometric fuzzy Korovkin theory, fuzzy modulus of continuity. 2000 Mathematics Subject Classi…cation. 26E50, 41A25, 41A36. 1

334

2

OKTAY DUM AN AND GEORGE A. ANASTASSIOU (r)

(r)

(r) Now denote the interval [u]r by [u(r) _ ; u+ ]; where u_ r 2 [0; 1]: Then, for u; v 2 RF ; de…ne

u

v , u(r) _

(r)

(r)

u+ and u(r) _ ; u+ 2 R for

(r)

(r) v_ and u+

v+ for all 0

De…ne also the following metric D : RF RF ! R+ by n (r) (r) ; u+ v_ D(u; v) = sup max u(r) _

r (r)

v+

r2[0;1]

1: o

:

In this case, (RF ; D) is a complete metric space (see [12]). Let f; g : R ! RF be fuzzy number valued functions. Then, the distance between f and g on R is given by o n (r)

f_(r)

D (f; g) = sup sup max x2R r2[0;1]

(r) ; f+ g_

(r)

g+

:

In this article we consider that j ! 1: Let K be a subset of N. Then, the (asymptotic) density of K is de…ned by (K) := lim j

# fn

j : n 2 Kg j

provided the limit exists, where the symbol # fBg denotes the cardinality of a set B: Using this density Fast [13] introduced the notion of statistical convergence of number sequences as follows: (xn )n2N is statistically convergent to a number L if, for every " > 0; the set fn 2 N : jxn Lj "g has density zero, i.e., (fn 2 N : jxn

Lj

"g) = lim j

# fn

j : jxn j

Lj

"g

= 0:

Now let A = (ajn ) be an in…nite summability matrix. P1 Then, the A-transform of x, denoted Ax := ((Ax)j ); is given by (Ax)j = n=1 ajn xn ; provided the series converges for each j: We say that A is regular if limj (Ax)j = L whenever limj xj = L [14]. Assume now that A is a nonnegative regular summability matrix and K is a subset of N. The A-density of K is de…ned by X ajn A (K) := lim j

provided the limit exists. Observe matrix of order one, de…ned by 8 < cjn = :

n2K

that if we take A = C1 = (cjn ); the Cesáro 1 ; if 1 n j j 0; otherwise

then C1 (K) = (K) for any subset K of N. With the help of the A-density, Freedman and Sember [16] introduced the notion of A-statistical convergence, which is a more general method of statistical convergence. Recall that the sequence (xn )n2N is said to be A-statistically convergent to L if, for every " > 0; Lj "g = 0; or equivalently A fn 2 N : jxn X lim ajn = 0: j

n: jxn Lj "

This limit is denoted by stA limn xn = L: It is not hard to see that if we take A = C1 ; then C1 -statistical convergence coincides with the statistical convergence mentioned above. If A is replaced by the identity matrix, then we get the ordinary

335

STATISTICAL TRIGONOM ETRIC FUZZY APPROXIM ATION

3

convergence of number sequences. We also note that if A = (ajn ) is any nonnegative regular summability matrix for which limj maxn fajn g = 0; then A statistical convergence is stronger than convergence (see [17]). Actually, every convergent sequence is A-statistically convergent to the same value for any non-negative regular matrix A, but its converse is not always true. Some other results regarding statistical and A-statistical convergences may be found in the papers [18, 19]. Now let ( n )n2N be a fuzzy number valued sequence. Then, Nuray and Sava¸s [20] introduced the fuzzy analog of statistical convergence by using the fuzzy metric D instead of the classical absolute value in the above de…nition. So, by a similar idea, one can obtain the following de…nition of A-statistical convergence of fuzzy valued sequences. We say that ( n )n2N is A-statistically convergent to 2 RF ; which is denoted by stA limn D( n ; ) = 0; if for every " > 0; A (fn 2 N : D( n ; ) "g) = 0; i.e., X ajn = 0 lim j

n:D(

n;

) "

holds. Of course, the case of A = C1 immediately reduces to the statistical convergence of fuzzy valued sequences. Also, replacing A with the identity matrix, we get the classical fuzzy convergence introduced by Matloka [21]. 2. Statistical Fuzzy Trigonometric Korovkin Theory In this section we prove a fuzzy trigonometric Korovkin-type approximation theorem by means of A-statistical convergence. In order to show that our result is stronger than its classical case we display an example of fuzzy positive linear operators by using fuzzy Fejer operators. Let f : R ! RF be fuzzy number valued functions. Then f is said to be fuzzy continuous at x0 2 R provided that whenever xn ! x0 , then D (f (xn ); f (x0 )) ! 1 as n ! 1: Also, we say that f is fuzzy continuous on R if it is fuzzy continuous at every point x 2 R: The set of all fuzzy continuous functions on R is denoted by CF (R) (see, for instance, [7, 9]). Notice that CF (R) is only a cone not a vector space. (F ) By C2 (R) we mean the space of all fuzzy continuous and 2 -periodic functions on R. Also the space of all real valued continuous and 2 -periodic functions is denoted by C2 (R): Assume that f : [a; b] ! RF is a fuzzy number valued function. Then, f is said to be fuzzy-Riemann integrable (or, F R-integrable) to I 2 RF if, for given " > 0; there exists a > 0 such that, for any partition P = f[u; v]; g of [a; b] with the norms (P ) < ; we have ! M D (v u) f ( ); I < ": P

In this case, we write Zb I := (F R) f (x)dx: a

By Corollary 13.2 of [10, p. 644], we conclude that if f 2 CF [a; b] (fuzzy continuous on [a; b]), then f is F R-integrable on [a; b]:

336

4

OKTAY DUM AN AND GEORGE A. ANASTASSIOU

Now let L : CF (R) ! CF (R) be an operator. Then L is said to be fuzzy linear if, for every 1 ; 2 2 R, f1 ; f2 2 CF (R); and x 2 R; L(

1

f1

f2 ; x) =

2

L(f1 ; x)

1

2

L(f2 ; x)

holds. Also L is called fuzzy positive linear operator if it is fuzzy linear and, the condition L(f ; x) L(g; x) is satis…ed for any f; g 2 CF (R) and all x 2 R with f (x) g(x): Throughout the paper we use the test functions fi (i = 0; 1; 2) de…ned by f0 (x) = 1; f1 (x) = cos x; f2 (x) = sin x: Then, we get the following result. Theorem 2.1. Let A = (ajn ) be a non-negative regular summability matrix and let (F ) fLn gn2N be a sequence of fuzzy positive linear operators de…ned on C2 (R). As~ n gn2N of positive linear operators sume that there exists a corresponding sequence fL de…ned on C2 (R) with the property (2.1)

fLn (f ; x)g

(r)

~n f =L

(r)

;x

(F )

for all x 2 [a; b]; r 2 [0; 1]; n 2 N and f 2 C2 (R):Assume further that (2.2)

stA

~ n (fi ) lim L n

fi = 0 for each i = 0; 1; 2;

the symbol kgk denotes the usual supremum norm of g 2 C2 (R). Then, for all (F ) f 2 C2 (R); we have stA

lim D (Ln (f ); f ) = 0: n

Proof. Suppose that I is a closed bounded interval with hlength 2 of R. iNow let (r) (F ) f 2 C2 (R); x 2 I and r 2 [0; 1]: Taking [f (x)](r) = f_(r) (x); f+ (x) we get (r) _ Hence, for every " > 0; there exists a > 0 such that f 2 C2 (R): (2.3) for all y satisfying jy

f

(r)

(y)

f

(r)

(x) < "

xj < : On the other hand, by the boundedness of f

(2.4)

f

(r)

(y)

f

(r)

(x)

2 f

f

(r)

(y)

f

(r)

(x)

" + 2M

,

(r)

holds for all y 2 R. Now consider the subinterval (x 2 : Then, by (2.3) and (2.4), it is not hard to see that (2.5)

(r)

(r)

;2 + x

] with length

'(y) sin2 2 (r)

(r)

holds for all y 2 (x ;2 +x ]; where '(y) := sin2 y 2 x and M := f : Observe that inequality (2.5) also holds for all y 2 R because of the periodicity (r) of f (see, for instance, [15]). Now using the linearity and the positivity of the

337

STATISTICAL TRIGONOM ETRIC FUZZY APPROXIM ATION

5

~ n and considering inequality (2.5), we may write, for each n 2 N, that operators L ~ n f (r) ; x L

f

(r)

~n L

(x)

f

+M

(r)

(r)

(y)

f

(r)

~ n (f0 ; x) L

"+ "+M

(r)

(x) ; x f0 (x)

~ n (f0 ; x) L

f0 (x)

(r)

+

2M sin2

~ n ('; x) : L 2

Hence, we obtain that (r)

~ n f (r) ; x L

f

(r)

(x)

"+

"+M

(r)

2M + sin2

2

!

~ n (f0 ; x) L

f0 (x)

(r)

2M + 2 sin

~ n (f1 ; x) L

f1 (x)

2M sin2

~ n (f2 ; x) L

f2 (x) :

2 (r)

+

Setting K see that

(r)

(") := " + M

~ n f (r) L

(2.6)

(r)

f

2

(r)

+

2M sin2

(r)

and taking supremum over x 2 R; we easily

2

"+K

(r)

~ n (f0 ) (")f L

~ n (f1 ) + L

f1

f0

~ n (f2 ) + L

f2 g:

Now it follows from (2.6) that D (Ln (f ); f )

= =

sup D (Ln (f ; x); f (x)) x2R n ~ n f (r) ; x sup sup max L _ x2R r2[0;1]

=

sup max r2[0;1]

n

~ n f (r) L _

~ n f (r) ; x f_(r) (x) ; L +

~ n f (r) f_(r) ; L +

(r)

f+

Therefore, combining the above equality with (2.6), we have

(2.7)

D (Ln (f ); f )

~ n (f0 ) " + K(")f L ~ n (f2 ) + L

f2 g;

~ n (f1 ) f0 + L

f1

o

(r)

f+ (x) :

o

338

6

OKTAY DUM AN AND GEORGE A. ANASTASSIOU

n o (r) where K(") := sup max K_(r) ("); K+ (") . Now, for a given "0 > 0; chose " > 0 r2[0;1]

such that 0 < " < "0 ; and consider the following sets U

:

= fn 2 N : D (Ln (f ); f )

U0

:

=

~ n (f0 ) n2N: L

f0

U1

:

=

~ n (f1 ) n2N: L

f1

U2

:

=

~ n (f2 ) n2N: L

f2

"0 g ; "0 " 3K(") "0 " 3K(") "0 " 3K(")

; ; :

Then inequality (2.4) gives U

U0 [ U1 [ U2 ;

which guarantees that, for each j 2 N, X X X X ajn ajn + ajn + ajn : n2U

n2U0

n2U1

n2U2

If we take limit as j ! 1 on the both sides of inequality (2.6) and use the hypothesis (2.2), we immediately see that X lim ajn = 0; j

n2U

whence the result. Concluding Remarks. 1. If we replace the matrix A in Theorem 2.1 by the Cesáro matrix C1 ; we immediately get the following statistical fuzzy Korovkin result in the trigonometric case. Corollary 2.2. Let fLn gn2N be a sequence of fuzzy positive linear operators de(F ) ~ n gn2N be a corresponding sequence of positive linear …ned on C2 (R), and let fL operators de…ned on C2 (R) with the property (2:1). Assume that st

~ n (fi ) lim L n

fi = 0 for each i = 0; 1; 2:

(F )

Then, for all f 2 C2 (R); we have st

lim D (Ln (f ); f ) = 0: n

2. Replacing the matrix A by the identity matrix, one can obtain the classical fuzzy Korovkin result which was introduced by Anastassiou and Gal [9]. Corollary 2.3 ([9]). Let fLn gn2N be a sequence of fuzzy positive linear operators (F ) ~ n gn2N be a corresponding sequence of positive linde…ned on C2 (R), and let fL ear operators de…ned on C2 (R) with the property (2:1). Assume that the sequence ~ n (fi )gn2N is uniformly convergent to fi on the whole real line (in the ordinary fL (F ) sense). Then, for all f 2 C2 (R); the sequence fLn (f )gn2N is uniformly convergent to f on the whole real line (in the fuzzy sense).

339

STATISTICAL TRIGONOM ETRIC FUZZY APPROXIM ATION

7

3. Now the following application shows that our A-statistical fuzzy Korovkintype approximation theorem in the trigonometric case (Theorem 2.1) is a non-trivial generalization of its classical case (Corollary 2.3) given by Anastassiou and Gal [9]. Let A = (ajn ) be any non-negative regular summability matrix. Assume that K is any subset of N satisfying A (K) = 0: Then de…ne a sequence (un )n2N by: (2.8)

un =

p 0;

n; if n 2 K if n 2 NnK.

In this case, observe that (un )n2N is non-convergent (in the ordinary sense). However, since for every " > 0 X X lim ajn = lim ajn = A (K) = 0, j

j

n:jun j "

n2K

we have (2.9)

stA

lim un = 0; n

although the sequence (un )n2N is unbounded from above. Now de…ne the fuzzy Fejer operators Fn as follows: 8 9 Z 2 n < = sin 2 (y x) 1 (F R) f (y) (2.10) Fn (f ; x) = dy ; y x 2 : ; n 2 sin 2 (F )

where n 2 N, f 2 C2 (R) and x 2 R. Then observe that the operators Fn are fuzzy positive linear. Also, the corresponding real Fejer operators have the following form: Z sin2 n2 (y x) 1 (r) (r) fFn (f ; x)g = F~n f ; x := dy f (y) n 2 sin2 y 2 x

where f

(r)

2 C2 (R) and r 2 [0; 1]: Then, we obtain that (see [15]) F~n (f0 ; x)

=

F~n (f1 ; x)

=

F~n (f2 ; x)

=

1; n

1 n

n

1 n

cos x; sin x:

Now using the sequence (un )n2N given by (2.8) we introduce the following fuzzy (F ) positive linear operators de…ned on the space C2 (R) : (2.11)

Tn (f ; x) = (1 + un )

Fn (f ; x);

(F )

where n 2 N, f 2 C2 (R) and x 2 R. So, the corresponding real positive linear operators are given by 1 + un (r) T~n f ; x := n

Z

f (y)

sin2

n x) 2 (y y x 2 2 sin 2

dy;

340

8

where f

OKTAY DUM AN AND GEORGE A. ANASTASSIOU (r)

2 C2 (R): Then we get, for all n 2 N and x 2 R, that T~n (f0 )

f0

T~n (f1 )

f1

T~n (f2 )

f2

= un ; 1 + un ; n 1 + un un + : n un +

It follows from (2.9) that (2.12)

lim T~n (f0 )

stA

n

f0 = 0:

Also, by the de…nition of (un )n2N we have lim n

1 + un = 0; n

which yields, for any non-negative regular matrix A = (ajn ); that (2.13)

stA

lim n

1 + un = 0: n

Now by (2.9) and (2.13) we easily see that, for every " > 0; X X X lim ajn lim ajn + lim j j j n n:jun j 2" n:kT~n (f1 ) f1 k " n:j 1+u j n

ajn = 0: " 2

So we get (2.14)

stA

lim T~n (f1 ) n

f1 = 0:

By a similar idea, one can obtain that (2.15)

stA

lim T~n (f2 ) n

f2 = 0:

Now, with the help of (2.12), (2.14), (2.15), all hypotheses of Theorem 2.1 hold. (F ) Then, we conclude, for all f 2 C2 (R), that stA

lim D (Tn (f ); f ) = 0: n

However, since the sequence (un )n2N is non-convergent and also unbounded from above, the sequence fTn (f )gn2N is not fuzzy convergent to f: Hence, Corollary 2.3 does not work for the operators Tn de…ned by (2.11). 3. Statistical Fuzzy Rates In the classical summability settings rates of summation have been introduced in several ways (see for instance, [22, 23, 24]). The concept of statistical rates of convergence, for nonvanishing two null sequences, is studied in [23]. Furthermore, various ways of de…ning rates of convergence in the A-statistical sense have been introduced in [3] as the following way. Let A = (ajn ) be a non-negative regular summability matrix and let (pn )n2N be a positive non-increasing sequence of real numbers. Then

341

STATISTICAL TRIGONOM ETRIC FUZZY APPROXIM ATION

9

(a) A sequence x = (xn ) is A-statistically convergent to the number L with the rate of o(pn ) if for every " > 0; X 1 lim ajn = 0: j pj n:jxn Lj "

In this case we write xn (b) If for every " > 0;

L = stA

sup j

1 pj

X

n:jxn j "

o(pn ) as n ! 1: ajn < 1;

then (xn )n2N is A statistically bounded with the rate of O(pn ) and it is denoted by xn = stA O(pn ) as n ! 1: (c) (xn )n2N is A-statistically convergent to L with the rate of om (pn ), denoted by xn L = stA om (pn ) as n ! 1; if for every " > 0; X ajn = 0: lim j

n:jxn Lj "pn

(d) (xn )n2N is A-statistically bounded with the rate of Om (pn ) provided that there is a positive number M satisfying X ajn = 0; lim j

n:jxn j M pn

which is denoted by xn = stA

Om (pn ) as n ! 1:

Notice that, in de…nitions (a) and (b); the “rate”is more controlled by the entries of the summability method rather than the terms of the sequence (xn )n2N . But, in order to see the e¤ect on the terms of the sequence we need the de…nitions (c) and (d), respectively. We should also remark that, for the convergence of fuzzy number valued sequences or fuzzy number valued function sequences, we have to use the metrics D and D instead of the absolute value metric in all de…nitions mentioned above. (F ) Let f 2 C2 (R). Then the (…rst) fuzzy modulus of continuity of f , which is introduced by [10] (see also [7, 9]), is de…ned by (F )

w1 (f; ) := for any

sup

D (f (x); f (y))

x;y2R; jx yj

> 0: With this terminology, we have the following result.

Theorem 3.1. Let A = (ajn ) be a non-negative regular summability matrix and let (F ) fLn gn2N be a sequence of fuzzy positive linear operators de…ned on C2 (R): Assume ~ n gn2N of positive linear operators on that there exists a corresponding sequence fL C2 (R) with the property (2:1). Suppose that (an )n2N and (bn )n2N are positive non~ n satisfy the following conditions: increasing sequences and also that the operators L (i)

~ n (f0 ) L (F )

(ii) w1 (f; sin2

y x 2

f0 = stA n ) = stA

o(an ) as n ! 1; o(bn ) as n ! 1; where

for each x 2 R:

n

=

r

~ n (') and '(y) = L

342

10

OKTAY DUM AN AND GEORGE A. ANASTASSIOU (F )

Then, for all f 2 C2 (R); we have D (Ln (f ); f ) = stA

o(cn ) as n ! 1,

where cn := maxfan ; bn g for each n 2 N. Furthermore, similar results hold when little \o" is replaced by big \O". (F )

Proof. Let f 2 C2 (R): Then, using the property (2.1) and applying Theorem 4 of [9], we immediately see, for each n 2 N, that ~ n (f0 ) M L

D (Ln (f ); f )

~ n (f0 ) + f0 w(F ) (f; f0 + L 1

n );

where M := D f; f0g and f0g denotes the neutral element for : It follows from the above inequality that (3.1) ~ n (f0 ) f0 + L ~ n (f0 ) f0 w(F ) (f; n ) + 2w(F ) (f; n ) D (Ln (f ); f ) M L 1 1 holds for each n 2 N. Now, for a given " > 0; consider the following sets: V V0 V1 V2 V3

:

= fn 2 N : D (Ln (f ); f ) n ~ n (f0 ) f0 : = n2N: L

"g ; " o ; 3M r " ~ n (f0 ) f0 : = n2N: L ; 3 r " (F ) : = n 2 N : w1 (f; n ) ; 3 n o " (F ) : = n 2 N : w1 (f; n ) : 6

Hence, inequality (3.1) implies that V V0 [ V1 [ V2 [ V3 : Then we may write, for each j 2 N, that 1 X 1 X 1 X 1 X 1 X (3.2) ajn ajn + ajn + ajn + ajn : cj cj cj cj cj 0 00 n2V

n2V0

n2V1

n2V1

n2V2

Also using the fact cj = maxfaj ; bj g; we obtain from (3.2) that 1 X 1 X 1 X 1 X 1 X (3.3) ajn ajn + ajn + ajn + ajn : cj aj aj bj bj 0 00 n2V

n2V0

n2V1

n2V1

n2V2

Therefore, letting j ! 1 on the both sides of inequality (3.3) and using the hypotheses (i) and (ii), we conclude that 1 X lim ajn = 0; j cj n2V

which means that stA for all f 2

lim D (Ln (f ); f ) = 0 n

(F ) C2 (R):

The above proof can easily be modi…ed to prove the following analog.

343

STATISTICAL TRIGONOM ETRIC FUZZY APPROXIM ATION

11

Theorem 3.2. Let A = (ajn ) be a non-negative regular summability matrix and (F ) let fLn gn2N be a sequence of fuzzy positive linear operators on C2 (R). Assume ~ n gn2N of positive linear operators on that there exists a corresponding sequence fL C2 (R) with the property (2:1). Suppose that (an )n2N and (bn )n2N are positive non~ n satisfy the following conditions: increasing sequences and also that the operators L ~ n (f0 ) f0 = stA om (an ) as n ! 1; (i) L (F )

(ii) w1 (f; n ) = stA om (bn ) as n ! 1; where 3:1. (F ) Then, for all f 2 C2 (R); we have D (Ln (f ); f ) = stA

n

is given as in Theorem

o(dn ) as n ! 1,

where dn := maxfan ; bn ; an bn g for each n 2 N. Furthermore, similar results hold when little \om " is replaced by big \Om ".

References [1] O. Duman, Statistical approximation for periodic functions, Demonstratio Math. 36 (2003) 873-878. [2] O. Duman and E. Erku¸s, Approximation of continuous periodic functions via statistical convergence, Comput. Math. Appl. 52 (2006) 967-974. [3] O. Duman, M.K. Khan, C. Orhan, A-Statistical convergence of approximating operators, Math. Inequal. Appl. 6 (2003) 689-699. [4] E. Erku¸s and O. Duman, A-Statistical extension of the Korovkin type approximation theorem, Proc. Indian Acad. Sci. Math. Sci. 115 (2005) 499-508. [5] A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002) 129-138. [6] G.A. Anastassiou, Fuzzy random Korovkin theory and inequalities, Math. Inequal. Appl. 10 (2007) 63-94. [7] G.A. Anastassiou, On basic fuzzy Korovkin theory, Studia Univ. Babe¸s-Bolyai Math. 50 (2005) 3-10. [8] G.A. Anastassiou, Fuzzy approximation by fuzzy convolution type operators, Comput. Math. Appl. 48 (2004) 1369-1386. [9] G.A. Anastassiou and S.G. Gal, On fuzzy trigonometric Korovkin theory, Nonlinear Funct. Anal. Appl. 11 (2006) 385-395. [10] S.G. Gal, Approximation theory in fuzzy setting, Handbook of Analytic-Computational Methods in Applied Mathematics, 617–666, Chapman & Hall/CRC, Boca Raton, FL, 2000. [11] R.J. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18 (1986) 31–43. [12] C.X. Wu, M. Ma, Embedding problem of fuzzy number space I, Fuzzy Sets and Systems 44 (1991) 33–38. [13] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241-244. [14] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, UK, 2000. [15] P.P. Korovkin, Linear Operators and Theory of Approximation, Hindustan Publ. Corp., Delhi, 1960. [16] A.R. Freedman, J.J. Sember, Densities and summability, Paci…c J. Math. 95 (1981) 293-305. [17] E. Kolk, Matrix summability of statistically convergent sequences, Analysis 13 (1993) 77-83. [18] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301-313. [19] H.I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995) 1811-1819. [20] F. Nuray, E. Sava¸s, Statistical convergence of sequences of fuzzy numbers, Math. Slovaca 45 (1995) 269-273. [21] M. Matloka, Sequences of fuzzy numbers, BUSEFAL 28 (1986) 28-37.

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[22] J.A. Fridy, Minimal rates of summability, Canad. J. Math. 30 (1978) 808-816. [23] J.A. Fridy, H.I. Miller and C. Orhan, Statistical rates of convergence, Acta Sci. Math. (Szeged) 69 (2003) 147-157. [24] H.I. Miller, Rates of convergence and topics in summability theory, Akad. Nauka Umjet. Bosne Hercegov. Rad. Odjelj. Prirod. Mat. Nauka 22 (1983) 39-55.

Oktay Duman TOBB Economics and Technology University, Faculty of Arts and Sciences, Department of Mathematics, Sö¼ gütözü TR-06530, Ankara, TURKEY E-Mail: [email protected] George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA E-Mail: [email protected]

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.3,345-354,2008, COPYRIGHT 2008 EUDOXUS PRESS, 345 LLC

THE LAGRANGE - POINCARE´ EQUATIONS FOR A REFINEMENT OF A PRINCIPAL GP (n; R) - BUNDLE GHEORGHE IVAN, MIHAI IVAN and DUMITRU OPRIS¸ ABSTRACT. In this paper the geometric structures defined on the refinement of a principal bundle determined by the factorization of the projective group are studied. For this refinement the Poisson structure, the Lagrange - Poincar´ e and Wong equations are written. 1

Introduction The paper consists of three sections. In the first section we present some definitions and results developed in [2] concerning the geometry of the manifold T Q/G. The manifold T Q/G is isomorphic with T (Q/G) ⊕ GeG by the isomorphism αAG associated to a connection AG . The brackets on the vector fields of X (Q/G) ⊕ Sect(GeG ) and the Poisson structure on T ∗ (Q) ⊕ GeG are presented. We associate the reduced of a G invariant Lagrangian for which the Lagrange - Poincar´ e equations are written. The second section deals with refinements of a differentiable principal bundle defined by closed subgroups of the structure group. Also, we define the reduced bundles associated to a refinement of a principal bundle. In the Section 3 is used the theory of reduction described in the first section for the fibre bundles which constitute a refinement of a principal bundle having the projective group as structure group. Throughout the paper all manifolds are of finite dimension and whitout boundary. All maps are differentiable of C ∞ - class. We use the definitions and results about the fibre bundles and connections defined on manifolds given in [1], [4] and [5]. Notations used are the same as those in [2] and [6]. 1. The reduced bundle of a principal G - bundle and the reduced Lagrangian. We assume that we have the following set up: a manifold Q and an action of the Lie group G on Q, say ρ : G × Q → Q. If πG : Q → Q/G is a left principal bundle and the Lie group G acts differentiably on the manifold F on the left, then the associated fibre bundle with standard fibre F is, by definition, Q ×G F = (Q × F )/G, where the action of G on Q × F is given by a(q, y) = (aq, ay), (∀) q ∈ Q, y ∈ F, a ∈ G. Also, πF : Q ×G F → Q/G is a ( left ) fibre bundle with structure group G. Let G the Lie algebra of the Lie group G. The associated bundle with standard fibre G, where the action of G on G is the adjoint action is called the adjoint bundle ( here 1

2000 Mathematics Subject Classification : 55R05,53C05, 58A30. Key words and phrases: adjoint bundle, invariant bundle, connection, refinement

1

346

IVAN ET AL

eG : Ge → Q/G F = G, ρg = Adg , g ∈ G ) and it is denoted by GeG = AdG (Q). We let π eG ([q, ξ]G = [q]G . Each fibre GexG of GeG carries a natural denote the projection given by π Lie algebra structure defined by [ [q, ξ]G , [q, η]G ] = [ q, [ξ, η]G ]. Let T Q be the tangent bundle on Q. An element of Tq Q will be denoted by vq , uq , . . . or by (q, q). ˙ The tangent lift of the action of G on Q defines an action of G on T Q and we can form the quotient (T Q)/G =: T Q/G. There is a well defined map τQ : T Q/G → Q/G induced by the tangent of the projection map πG : Q → Q/G and given by [vq ]G → [q]G . The rules [vq ] + [uq ]G = [vq + uq ]G and λ[vq ]G = [λvq ]G , where λ ∈ R, vq , uq ∈ Tq Q and [vq ]G and [uq ]G are their equivalence classes in the quotient T Q/G define a vector bundle structure on T Q/G having the base Q/G. The fibre (T Q/G)x is isomorphic, as vector space, to Tq Q, for each x = [q]G . Let a connection AG on Q given by one form AG : T Q → G with the properties:

(1) AG (ξq) = ξ, for all ξ ∈ G; (2) AG (Tq ρg · v) = Adg (AG (v)), where Adg is the adjoint action of G on G. The restriction of a connection to Tq Q is denoted AG q and the vertical and horizontal space defined at q ∈ Q is V erq = KerTq πG , Horq = AG q . G G A The curvature of A , denoted by B is a Lie algebra valued two form on Q given by: G

B A (uq , vq ) = dAG (Horq (uq ), Horq (vq )). We assume that we have a connection AG on the bundle πG : Q → Q/G. The map αAG : T Q/G → T (Q/G) ⊕ GeG defined by αAG ([q, q] ˙ G ) = T πG (q, q) ˙ ⊕ [q, AG (q, q)] ˙ G is a well defined vector bundle isomorphism. Let T Q = Hor(T Q) ⊕ V er(T Q) be the decomposition into horizontal and vertical parts. Since the bundles Hor(T Q) and V er(T Q) are G- invariant we have T Q/G = Hor(T Q)/G ⊕ V er(T Q)/G. We have that αAG (V er(T Q)/G) = T (Q/G) and αAG (V er(T Q)/G) = GeG . Let ιG (T Q) : IG (T Q) → Q/G be the vector bundle whose fibre (ιG (T Q))−1 (x) at an element x = [q]G ∈ Q/G is the vector space of all invariant vector fields on Q along −1 πG . That is, an element of IG (T Q) is a vector field, say, Z, defined only at points −1 −1 q ∈ πG (x), that is Z(q) ∈ T Q for all q ∈ πG (x) such that gZ = Z, (∀) g ∈ G. H (T Q) → Q/G V V We also let ιG (T Q) : IG (T Q) → Q/G ( resp., ιG (T Q)H : IG V −1 ) be the vector bundle whose fibre (ιG (T Q) ) (x) ( resp., (ιG (T Q)H )−1 (x) ) at an element x = [q]G ∈ Q/G is the vector space of all vertical ( resp., horizontal ) invariant −1 V (T Q) ( resp., I H (T Q) ) is a vector vector fields on πG . That is, an element of IG G 2

LAGRANGE-POINCARE EQUATIONS

347

−1 field, say, Y ( resp., X ) on the manifold πG (x) such that gY = Y, (∀) g ∈ G V (T Q) → Q/G ( resp., ( resp., gX = X, (∀) g ∈ G ). We call ιG (T Q)V : IG H V ιG (T Q) : IG (T Q) → Q/G ) the vertical ( resp., horizontal ) invariant bundle. H (T Q)) and Sect(I V (T Q)) the Lie algebra of sections of the horizontal Let Sect(IG G and vertical invariant bundle, respectively. The map T πG establishes a well defined isoH (T Q)) and X (Q/G). We can deduce that there are natural morphism between Sect(IG identification Sect(T (Q/G) ⊕ GeG ) = X (Q/G) ⊕ Sect(GeG ). a } are the Given the basis {εa | a = 1, p } for the Lie algebra G for which {Cbc ∂ structure constants, we obtain the local basis { , ea } for πG : T Q/G → Q/G and ∂xi V c e . { ea } for IG (T Q) → Q/G such that [ea , eb ] = Cab c Let { Aai (x) } the local functions on Q/G for the connection AG defined for e AG ξ of a section ξ = ξ a ea πG : Q → Q/G. The corresponding covariant derivative ∇ G V (T Q) reads ∇ e A ξ : Q/G → T ∗ (Q/G) ⊕ I V (T Q), of IG G G

eA ξ = ( ∇

∂ξ a a Gb c + Cbc Ai ξ )dxi ⊗ ea ∂xi

e AG ξ is given by and if X ∈ X (Q/G), then ∇ X G

e A ξ = X i( ∇ X

∂ξ a a Gb c + Cbc Ai ξ )ea . ∂xi

In particular, we have (1.1)

G

e A ea = C c AGb ec . ∇ i ba i

e Ga dxi ∧ dxj ⊗ ea , where e AG is given by B e AG = 1 B The curvature B 2 ij

(1.2)

e Ga = B ij

∂AGa AGa j a Gb Gc i − + Cbc Ai Aj . ∂xi ∂xj

Let Xi ⊕ ξ i ∈ Sect(T (Q/G) ⊕ GeG ), i = 1, 2 be given two sections. Then e AG ξ2 − ∇ e AG ξ1 − B e AG (X1 , X2 ) + [ξ , ξ ]. [X1 ⊕ ξ 1 , X2 ⊕ ξ 2 ] = [X1 , X2 ] ⊕ ∇ 1 2 X1 X2 ∂ ⊕ ea , i = 1.n, a = 1, p } we have For { ∂xi ∂ ∂ d Gc d Gc e AG d + C d )ed . (1.3) [ i ⊕ ea , j ⊕ eb ] = (Ccb Ai − Cca Aj − B ij ab ∂x ∂x Let (xi , x˙ i , ξ a ) the local coordinates of T Q ⊕ GeG and (xi , pi , µa ) the local coordie ∗ is given by nates of T ∗ Q ⊕ Ge∗ . The structure Poisson on T ∗ Q ⊕ G

3

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∂ ∂ e AG c µc ∂ ∧ ∂ − ∧ −B ij i ∂q ∂pi ∂pi ∂µj ∂ ∂ ∂ ∂ d c −Cca µd AGc ∧ + Cab µc ∧ . i ∂pi ∂µa ∂µa ∂µb Let L : T Q → R be an invariant Lagrangian, that is L(g(q, q)) ˙ = L(q, q), ˙ for all (q, q) ˙ ∈ T Q, g ∈ G. Because holds this invariance, we get a well defined reduced Lagrangian lG : T Q/G → R satisfying the relation lG ([q, q] ˙ G ) = L(q, q). ˙ We will consider lG as a function defined on T (Q/G) ⊕ GeG or T Q/G interchangeably, using the isomorphism αAG . Also, we will write lG (qG , q˙G , ξ), to emphasize the dependence of lG on (qG , q˙G ) ∈ T (Q/G) and ξ ∈ GeG . The vertical resp., horizontal Lagrange-Poincar´ e equation for lG is given by (1.5.1) resp., (1.5.2):

(1.4)

ΛG =

G

(1.5.1)

DA ∂lG ∂lG (qG , q˙G , ξ) = ad∗ξ (qG , q˙G , ξ), Dt ∂ξ ∂ξ

(1.5.2)

∂lG DA ∂lG (qg , q˙G , ξ) − (qG , q˙G , ξ) = ∂qG Dt ∂ q˙G

ξ ∈ GeG

G

∂lG e AG (qG ) >, ξ ∈ GeG . (qG , q˙G , ξ, iq˙G B ∂ξ In local coordinates the equation (1.5.1) resp., (1.5.2) becomes : =
(x, x, ˙ ξ) − (x, x, ˙ ξ) =< (x, x, ˙ ξ), ix˙ B ∂x Dt ∂ x˙ ∂ξ

The equations (1.10.1) and (1.10.2) are the first and the second Wong’s equation. Locally, the expression of the Lagrangian lG is the following: 1 1 a b (1.11) lG (x, x, ˙ ξ) = κG ˙ i x˙ j . ab ξ ξ + gij (x)x 2 2 The local expression of the vertical resp., horizontal Lagrange-Poincar´ e equation is given by (1.12.1) resp., (1.12.2): (1.12.1)

dpb a Gd i = −pa Cdb Ai x˙ , dt

(1.12.2)

1 ∂g jk dpi Ga j = −pa Bji x˙ − pj pk , dt 2 ∂xi

where pa =

∂lG = κab ξ b ∂ξ a

where pi = gij xj .

2. The reduced bundles associated to a refinement of a principal G - bundle. Let G be a Lie group and Nq = (G = H0 ⊃ H1 ⊃ . . . ⊃ Hq−1 ⊃ Hq = {e}) ( e is the identity element of G ) a sequence of Lie groups such that Hj is a closed subgroup of Hj−1 for 1 ≤ j ≤ q. Let (ξ, Nq ) be a structure consisting of a differentiable principal G - bundle ξ = (E, p, B, G) and Nq a sequence of closed subgroups of G. Let Ej = E/Hj , j = 0, q, Hkj = Hj /Hk and Gkj = Hj /Njk for 0 ≤ j < k ≤ q, where Njk is the largest normal subgroup of Kj included in Hk and Hj /Njk is the factor group of Hj by Njk . Finally, let pjk : zHk ∈ Ek → zHj ∈ Ej , (∀)z ∈ E for 0 ≤ i < j ≤ 2, the canonical map. The pair (ξ, Nq ) defines the fibre bundles ( see [6], MR 53 # 4058 ): ξjk = (Ek , pjk , Ej , Hkj , Gjk ) , 0 ≤ j < k ≤ q. The triplet (ξ; ξ0j , ξjq ), where ξ0j = (Ej , p0j , B, Hj0 , G0j ), ξjq = (E, pjq , Ej , Hj ) is called the refinement of ξ defined by Hj . We have that ξjq is a principal Hj - bundle for all 0 < j < q. When q = 2 and H1 = H, the refinement of ξ defined by H is the triple (ξ; ξ01 , ξ12 ) with ξ01 = (E/H, p01 , B, G/H, G/N ) and ξ12 = (E, p12 , E/H, H), where N is the largest normal subgroup of G included in H. This structure can be find in [4]. EXAMPLE 2.1. ([6]) Let ξ ∗ = (Ln M, p = πL , M, GL(n; R) be the principal bundle of tangent linear frames to n - manifold M. We consider the sequence N2∗ = (G = GL(n; R) ⊃ H = GT (n; R) ⊃ {e}), where GT (n; R) = { (aij ) ∈ GL(n; R) | aij = 0 for i > j } is the subgroup of upper triangular matrices. 5

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∗ , ξ ∗ ), ∗ = The refinement of ξ ∗ defined by GT (n; R) is (ξ ∗ ; ξ01 where ξ01 12 (Dn M, p01 , M, GL(n; R)/GT (n; R), GP (n − 1; R)) is the fibre bundle of tangent flags ∗ = (L M, p , D M, GT (n; R)). Here D M = L M/GT (n; R) is the to M and ξ12 n 12 n n n manifold of tangent flags to M and GP (n − 1; R)) = GL(n; R)/D(n; R) is the real projective group of order n−1, since the largest subgroup normal subgroup of GL(n; R) included in GT (n; R) is D(n; R) = { (λδji ) | (∀) λ ∈ R }. 2 REMARK 2.1. (i) If H is a closed subgroup of G such that N = {e}, then the refinement of ξ = (E, p, B, G) defined by H is (ξ; ξ01 , ξ12 ), where ξ01 = (E/H, p01 , B, G/H, G) and ξ12 = (E, p12 , E/H, H). (ii) If H is a normal closed subgroup of G, then the refinement of ξ = (E, p, B, G) defined by H is a refinement (ξ; ξ01 , ξ12 ), where ξ01 = (E/H, p01 , B, G/H) and ξ12 = (E, p12 , E/H, H) are principal bundles .2 EXAMPLE 2.2. Let M be a n - manifold and (T (M ), πT , M, Rn , GL(n; R)) be the fibre bundle of tangent vectors on M. By affine frame at x ∈ M we mean a triple u = (x, y, z), where (y, z) ∈ πT−1 (x) × −1 πL (x). We denote by An M the set of affine frames on M endowed with the differentiable structure canonically induced from the differentiable structure of M. In a local chart of M, an affine frame at x ∈ M is given by: ∂ ∂ (2.1) y = y i ( i )x , zj = zji ( i )x , det(zji ) 6= 0 ∂x ∂x and the local coordinates of u = (x, y, z) ∈ An M are (xi , y i , zji ), i, j = 1, n. µ ¶ 1 0 Let be the affine group GA(n; R) = { i ∈ GL(n + 1; R) | det(gji ) 6= 0 }. a gji The action of the Lie group G = GA(n; R) on the manifold An M is defined by the right translations τ(a,g) : G × An M → An M, (∀) (a, g) ∈ G with a = (ai ) ∈ Rn , g = (gji ) ∈ GL(n; R) where:

(2.2) ((a, g), u) → (τ(a,g) (u) = (x, y + za, zg) for all u = (x, y, z) ∈ An M. Let η = (An M, pA , M, GA(n; R)) be the principal bundle of affine frames to an n - manifold M, where pA : AN M → M is the canonical projection given by pA (u) = x, (∀) u = (x, y, z) ∈ An M. the sequence N21 = (GA(n; R) ⊃ µ We consider ¶ 1 0 T (n; R) ⊃ {e}), where T (n; R) = { i }. Since T (n; R) is normal in G, applya δji ing Remark 2.1.(ii), the refinement of η defined by T (n; R) is (η; η01 , η12 ), where η01 = (An M/T (n; R), p01 , M, GA(n; R)/T (n; R)) and η12 = (An M, p212 , An M/T (n; R), T (n; R)). 2 Let Q be a manifold and G a Lie group which acts differentiably on Q. We oonsider πG : Q → Q/G the principal bundle with the structure group G. We assume that is given a sequence N2 = (G ⊃ K ⊃ {e}) of closed subgroups of G. If we denote η = (Q, πG , Q/G, G), then the pair (η; N2 ) determines a refinement (η; η01 , η12 ) of η defined by K, where η01 = (Q/K, πGK , Q/G, G/K, G/N ) and η12 = (Q, πK , Q/K, K), and N is the largest 6

LAGRANGE-POINCARE EQUATIONS

351

normal subgroup of G included in K. Let AG and AK two connections on Q given by the forms AG : T Q → G, AK : T Q → K, where G resp., K is the Lie algebra of G resp., K. e = AdK (Q) and the isomorphisms α G : Let the adjoint bundles Ge = AdG (Q), K A G eK. e T Q/G → T (Q/G) ⊕ G , αAK : T Q/K → T (Q/K) ⊕ K e K → Q/K are called The vector bundles T (Q/G) ⊕ GeG → Q/G and T (Q/K) ⊕ K the reduced bundles associated to refinement defined by (η; N2 ). 3. Geometric structures on the fibre bundles of a refinement of a principal GP (n; R) - bundle. Let us we apply the above considerations in the case the group G is the projective group GP (n; R) and K is the affine group GA(n; R). Let G = GL(n + 1, R)/D(n + 1, R) the projective group of order n. The class [aij ] determined by the matrix (aij ) is denoted by a. The subgroup K of G is determined by all classes [aij ] for which the matrix (aij ) satisfy the condition an+1 = 0, h = 1, n h and it may be identified of the affine group of order n. We obtain thus a sequence N2 = (G ⊃ K ⊃ {e}), where G = GP (n; R) and K = GA(n; R). A base for the Lie algebra G of G is { εij , εi , εj } and we have ilp q [εij , εlk ] = δkqj εp ,

ip q [εi , εj ] = −γjq εp ,

[εij , εk ] = δki εj , where

[εij , εk ] = −δjk εi ,

ilp δkqj = δki δql δjp − δjl δqi δkp ,

[εi , εj ] = 0. [εi , εj ] = 0, ip γjq = δji δqp + δqi δjp .

A base for the Lie algebra K of K is { εij , εj } and we have ilp q [εij , εlk ] = δkqj εp ,

[εij , εk ] = δki εj ,

[εi , εj ] = 0.

Let πG : Q → Q/G the principal bundle having the projective group G as structure n+i n+i i i i group. Let hn+i n+i (Un+i , χn+i ) the local charts of Q with the coordinates (q , xj , x , xj ). The base of sections of the vector bundle GeG → Q/G is eij = xhj

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , ej = xhj h − xj (xhk h + xh h + xk ), ei = xh h + . + xj h ∂xi ∂x ∂x ∂xk ∂xi ∂xi ∂xk ∂xi

Let AG a connection on the principal bundle πG : Q → Q/G given by the functions on Q/G. The following relations hold:

(Pjli , Pli , Pjl )

7

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IVAN ET AL

(3.1)

 e AG el = δ lps P r eq + γ lq P p eq + Pki el  ∇  qkr pi s pk i ∂ k     i  ∂q     e AG l l eh + γ ls P k ep , e AG ek = P j ej − γ sq Pqi ep ∇ ∂ e = Phi ∇ s pk i s ki pk ∂     ∂q i ∂q i       e AG = 1 B l dq i ∧ dq j ⊗ ek + 1 Blij dq i ∧ dq j ⊗ el + 1 B l dq i ∧ dq j ⊗ el .  B kij l ij

2 2 2 Let (q i , q˙i , ξkl , ξk , ξ l ) the local coordinates on T Q ⊕ GeG and (q i , pi , µlk , µk , µl ) the ∗ local coordinates on T ∗ Q⊕ GeG . The structure Poisson is given by the following relations: (3.2.1) {q i , q j } = 0, {q i , pj } = δji , {q i , µlk } = 0, {q i , µl } = 0, {q i , µl } = 0; l µk − B µl − B l µ ; (3.2.2) {pi , pj } = −Bkij lij ij l l

j {pi , µlk } = µlj Pik − µqp (δip Pqk + δql Pki );

l ; (3.2.3) {pi , µl } = µkp (δkp Pil + δkl Pip ) + µh Phi

{µij , µlk } = δki µlj − δjl µik ;

(3.2.4) {µij , µl } = −δjl µi , (3.2.5) {µl , µk } = 0,

{µl , µk } = −(δkl δqp + δql δkp )µqp ;

{µij , µl } = δli µj ,

{µl , µk } = 0.

Let κG the bi-invariant Riemannian metric on G and the Lagrangian L : T Q → R given by (1.8). The reduced Lagrangian lG on T (Q/G) ⊕ GeG is given by : (3.3)

1 1 ij i j 1 i k lG (q i , q˙i , ξji , ξj , ξ i ) = κij ξ i ξ j + κjl ik ξj ξl + κ ξ ξ + 2 2 2 1 1 i h i κ ξ ξ + κli ξl ξ i + gij q˙i q˙j . +κjil ξ l ξij + κjl ξ ξ + ih l j i 2 2

and we have (3.4)

 ∂lG ∂lG  jl k j l jl  i  = κ ξ + κ ξ + κ ξ , p = = κlih ξlh + κih ξ h + κli ξl i l  pj = l i ih il ∂ξi ∂ξji  ∂lG ∂lG    pi = = κilh ξlh + κih ξ h + κih ξh , pi = = gij q˙j . ∂ξi ∂q i

The Lagrange-Poincar´ e equation and Wong’s equations for the projective group are given by the relations (3.5.1)-(3.5.4): dpji dt dpi (3.5.2) dt dpi (3.5.3) dt dp (3.5.4) i dt

(3.5.1)

j h = −[(plh (δlj Pik − δih Plk + (δih Pkj + δij Pkh )ph + Pik pj ] · q k h = −[Pik ph − (δhl Pik + δkl Phi )phl ] · q k i = −[Pkh pk + (δkl Phi + δki Phl )pkl ]q˙h h h = −(plh Blij + pij Bij + pl Blij )q˙j −

8

1 ∂g jl p p. 2 ∂q i j l

LAGRANGE-POINCARE EQUATIONS

353

Let πK : Q → Q/K the principal bundle having the affine group K as structure group and the local coordinates (q i , ηi ) on Q/K. The base of sections of the vector e K → Q/K is ei = xh ∂ , ej = xh ∂ . bundle K j j j ∂xh ∂xhi Let AK a connection on the principal bundle πK : Q → Q/K given by the functions hr h h (Ahr k , A , Akr , Ar ) on Q/K. The following relations (3.6) hold:  e AK el = (Apr δ l − Alr δ p )eq − Alr ek e AK el = (Ap δ l − Al δ p )eq − Al ek , ∇  ∇  qr k p r q k p k kr q k q  ∂ k ∂     r  ∂q ∂ηr    K K  ir A i A e e  ∇ ∇   ∂ ek = Akr ei , ∂ ek = (Ak ei 

(3.6)

r

∂q ∂ηr      e AK = 1 (B l dq i ∧ dq j + B lh dq i ∧ dηh + B lhi dηh ∧ dηi ) ⊗ ek +  B  ki k l   2 kij   1    + (B l dq i ∧ dq j + Bilh dq i ∧ dηh + B lhi dηh ∧ dηi ) ⊗ el .   2 ij

e K and (q i , ηi , pi , λi , µl , µl ) Let (q i , ηi , q˙i , η˙ i , ξkl , ξ l ) the local coordinates on T Q ⊕ K k ∗ ∗ K e the local coordinates on T Q ⊕ K . The structure Poisson is given by the relations: 1 l k l {q i , pj } = δij , {ηi , λj } = δij , {pi , pj } = − (Bkij µl + Bij µl ); 2 1 1 lj k µl + Bilj µl ), {λi , λj } = − (Bklji µkl + B lji µl ); {pi , λj } = − (Bki 2 2 p l p l q l l r l l lp r lr {pi , µk } = µp (Aki δq − Aqi δk ) − µk Ai ; {λ , µk } = µqp (Apr k δq − Aq δk ) − µk A ; i l i l l i i i {pr , µk } = Aikr µi , {λr , µk } = Air k µi ; {µj , µk } = −δk µj + δj µk , {µk , µj } = δk µj .

Let κK the bi-invariant Riemannian metric on K and the Lagrangian L : T Q → R e K is given by: given by (1.8). The reduced Lagrangian lK on T (Q/K) ⊕ K 1 1 lK (q i , ηi , q˙i , η˙ i , ξba , ξ a ) = κab ξ a ξ b + κcba ξ a ξcb + 2 2 1 1 1 1 ξ a ξ b + gij q˙i q˙j + gij q˙i ηj + g ij ηi ηj . + κcd 2 ab c d 2 2 2 and we have :   ∂lK ∂lK  bc d b c b  = κ ξ + κ ξ , p = = κcad ξcd + κab ξ b  pa = a ad c ac ∂ξba ∂ξ a (3.8)  ∂lK ∂lK   = gji q˙j + g ij η˙ j , = gij q˙j + gij η˙ j . pi =  pi = ∂ η˙ i ∂ q˙i (3.7)

The Lagrange-Poincar´ e equations and Wong’s equations for the affine group are given 9

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by the relations (3.9.1)-(3.9.4): dpba p bi p bi (3.9.1) = −[pqp (Abqi δqp − Abqi δap )q˙i − Abi pa q˙i + ppq (Api a δq − Aq δa ) · ηi − A pa ηi ] dt (3.9.2) (3.9.3)

dpa = −[pb Abai q˙i + pb Abi a η˙ i ] dt dpi ai h = −(pba Bbahi η˙ h + pba Bhai q˙h + pa B ahi η˙ h + pba Bbh q˙ )− dt −

(3.9.4)

∂ gej 1 ∂ gejl j l 1 ∂ gejl pj pl − l pj pl − p p 2 ∂ηi ∂ηi 2 ∂ηi

dpi aj a j a i = −(pba Bbji q˙ + pba Bbi η˙ j + pa Bji q˙ + pa Biaj η˙ j )− dt −

∂ gelj 1 ∂ gejl 1 ∂ gejl j l p p − pj pl − p p, j l i i 2 ∂q ∂q 2 ∂q i

where gejl are the elements of the inverse of matrix (gij ). References [1] R. Abraham, J.E. Marsden, T.S. Rat¸iu, Manifolds, Tensor Analysis and Applications.Second Edition. Mathematical Sciences 75, Springer - Verlag (1988). [2] H. Cendra, J.E. Marsden, T.S. Rat¸iu, Lagrangian Reduction by Stages. Mem. Amer.Math. Soc. 152,no.722(2001), 1-108. [3] Gh. Ivan,D.Opri¸s Old and new aspects in the study of refinements of a principal bundle. Tensor N.S.,63(2002), p.160 - 175. [4] S. Kobayashi, K. Nomizu,Foundations of differential geometry. Interscience Publ.,New York-London, Vol.I (1963). [5] L. Mangiarotti, G. Sardanashvily,Connections in classical and quantum field theory. World Scientific, Singapore, (2000). [6] D. I. Papuc, Sur les raffinements d’un espace fibr´ e principal diff´ erentiable. Anal. S¸t. Univ. ”Al. I. Cuza” Ia¸si, Sect. a I-a, Mat. ( N.S.), 18 (1972), p. 367-387. West University of Timi¸soara Seminarul de Geometrie ¸si Toplogie 4, Bd. V. Pˆarvan, 300223, Timi¸soara, Romania E-mail : [email protected],[email protected] and [email protected]

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.3,355-366,2008, COPYRIGHT 2008 EUDOXUS PRESS, 355 LLC

EXPLICIT p-ADIC q-EXPANSION FOR THE ALTERNATING SUMS OF POWERS

Leechae Jang and Taekyun Kim Department of Mathematics and Computer Science, KonKuk University, Chungju, S. Korea e-mail: [email protected], [email protected] EECS, Kyungpook National University, Taegu 702-701, S. Korea e-mail: [email protected] ( or [email protected]) Abstract.

In this paper, we give an explicit p-adic expansion of

X np

j=1 (j,p)=1

(−1)j [j]rq

as a power series in n. The coefficients are values of p-adic q-l-function for q-Euler numbers.

§1. Introduction Let p be a fixed prime. Throughout this paper Zp , Qp , C and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Qp , cf.[1, 4, 6, 14]. Let vp be the normalized exponential valuation of Cp with |p|p = p−vp (p) = p−1 . When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp . If q ∈ C, one normally assumes |q| < 1. If q ∈ Cp , Key words and phrases. p-adic q-integrals, Euler numbers, p-adic l-function. 2000 Mathematics Subject Classification: 11S80, 11B68, 11M99 . Typeset by AMS-TEX 1

356

2

L.C. JANG AND T. KIM 1

then we assume |q − 1|p < p− p−1 , so that q x = exp(x log q) for |x|p ≤ 1. Kubota and Leopoldt proved the existence of meromorphic functions, Lp (s, χ), defined over the p-adic number field, that serve as p-adic equivalents of the Dirichlet L-series, cf.[1, 14,15,16]. These p-adic L-functions interpolate the values 1 Lp (1 − n, χ) = − (1 − χn (p)pn−1 )Bn,χn , for n ∈ N = {1, 2, · · · , } , n where Bn,χ denote the nth generalized Bernoulli numbers associated with the primitive Dirichlet character χ, and χn = χw−n , with w the T eichm¨ uller character, cf.[8,14]. In [14,15], L. C. Washington have proved the following interesting formula: np X

   ∞  X −r −r 1 k 1−k−r = − (pn) L (r + k, w ), where is binomial coefficient. p jr k k k=1

j=1 (j,p)=1

To give the q-extension of the above Washington result, the author derived the sums of powers of consecutive q-integers as follows: (*)

n−1 X

q

l

[l]m−1 q

l=0

m−1   1 X m ml 1 = q βl [n]m−l + (q mn − 1)βm , see [7,8] , q m l m l=0

where βm are q-Bernoulli numbers. By using (*), we gave an explicit p-adic expansion np X j=1 (j,p)=1

 ∞  X −r qj =− [pn]kq Lp,q (r + k, w1−r−k ) [j]rq k

− (q − 1)

k=1

 ∞  X −r k=1

k

[pn]kq Tp,q (r

+ k, w

1−r−k

) − (q − 1)

p−1 X

(n) Bp,q (r, a : F ),

a=1

where Lp,q (s, χ) is p-adic q-L-function (see [7,12] ). Indeed, this is a q-extension result due to Washington, corresponding to the case q = 1, see [14]. Recently, the second author described algorithms to deal with nested symbolic sums over combinations of harmonic sum, binomial coefficients and denominators[13]. In addition, he treated Mellin transforms and the inverse Mellin transformation for functions that are encountered in Feynman diagram calculations. Together with results for the values of the higher harmonic sum at infinity the presented algorithm can be used for the symbolic

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p-ADIC q-EXPANSION FOR THE ALTERNATING SUMS OF POWERS

3

evaluation of whole classes of integrals that were thus far intractable [13]. The computation of Feynman diagrams has confronted physicists with classes of integrals that usually hard to be evaluated, both analytically and numerically. In [10], Rim and Kim treated explicit p-integral alternating harmonic sums. Harmonic sum is critical to explain the resonating phenomenon in a nature. It is also important as a solution of simple harmonic oscillating system in classical mechanics and quantum mechanics (see [17]). p-adic harmonic sum can be applied to these physical phenomena. With this application, p-adic harmonic sum also can be used for quantum statistical physics or quantum transportation theory( see [17]). Euler numbers and polynomials in alternating harmonic sum are used for Langevine equation of magnetism which is in the system with viscosity. For a fixed positive integer d with (p, d) = 1, set N X = Xd = lim ←− Z/dp Z, N [ ∗ X1 = Zp , X = a + dpZp , 0

a=1

χ(a)

 ∞  X −s j=0

j

q

ja



[F ]q [a]q

j Ej,qF , for s ∈ Zp .

This is a p-adic analytic function and has the following properties for χ = wt : (20)

lp,q (−n, wt ) = En,q − [p]nq En,qp , where n ≡ t ( mod p − 1),

(21)

lp,q (s, t) ∈ Zp for all s ∈ Zp when t ≡ 0( mod p − 1).

If t ≡ 0( mod p − 1), then lp,q (s1 , wt ) ≡ lp,q (s2 , wt )( mod p) for all s1 , s2 ∈ Zp , lp,q (k, wt ) ≡ lp,q (k + p, wt )( mod p). It is easy to see that (22)

      1 −r 1−r−k −1 −r k+j = , r+k−1 k j j+k k+j−1 j

for all positive integers r, j, k with j, k ≥ 0, j + k > 0, and r 6= 1 − k. Thus, we note that       1 −r 1−r−k 1 −r + 1 k + j (22-1) = . r+k−1 k j r−1 k+j j From (22) and (22-1), we derive (23)

      −r − 1 −r − k −r k+j r = . r+k k j k+j j

By using (13), we see that (24) n−1 n−1 X (−1)F l+a X
−r = (−1)l (−1)a [a]q + q a [F ]q [l]qF r [F l + a]q l=0 l=0  s   ∞ s−1   n X X [F ]q s nF l −r as a −r (−1) =− [a]q q (−1) q El,qF [n]s−l qF [a]q s 2 l s=1 l=0  s   ∞ Fs n X (1 − (−1)n )[a]−r [F ]q ) − 1) q as a −r ((−q −r [a]q q (−1) Es,qF + (−1)a . − [a]q s 2 2 s=1

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L.C. JANG AND T. KIM

For s ∈ Zp , we define the below T -Euler polynomials:  ∞  X
−s a −k ak a −s (25) Tn,q (s, a : F ) = (−1) < a > [ ]qF q (−1)n q nF k − 1 Ek,qF . k F k=1

Note that limq→1 Tn,q (s, a : F ) = 0, if n is even positive integer. From (23) and (24), we derive n−1 X

(−1)F l+a [F l + a]rq l=0  s  s−1   ∞  X (−q s )a X s nF l −r [F ]q −r =− [a]q q El,qF [n]s−l qF [a] 2 l s q s=1

(26)

l=0

−

w

−r

(a)

2

Tn,q (r, a : F ), where n is positive even integer .

Let n be positive even integer. Then, we evaluate the right side of Eq.(26) as follows: (27) s   s−1   ∞  X (−q s )a (−1)n X s nF l −r [F ]q −r q El,qF [n]s−l [a]q qF [a]q 2 l s s=1 l=0     l ∞ ∞  a X X r −r − 1 −r − k al [F ]q −k−r ak n k (−1) = [a]q q (−1) [F n]q q El,qF . r+k k 2 l [a]q k=1

l=0

It is easy to check that (28)

q

nF l

=

l   X l

j

j=0

[nF ]jq (q

j

− 1) = 1 +

l   X l j=1

j

[nF ]jq (q − 1)j .

Let (29)   l ∞  l   X X (−1)a −s al [F ]q l −s Kp,q (s, a : F ) =

q El,qF [nF ]jq (q − 1)j . 2 l [a]q j j=1 l=1

Note that limq→1 Kp,q (s, a; F ) = 0. For F = p, r ∈ N, we see that (30)

2

p−1 n−1 X X (−1)a+pl a=1 l=0

np X (−1)j = 2 . [a + pl]rq [j]rq j=1 (j,p)=1

365

p-ADIC q-EXPANSION FOR THE ALTERNATING SUMS OF POWERS

11

For s ∈ Zp , we define p-adic analytically continued function on Zp as Kp,q (s, χ) = 2

p−1 X

χ(a)Kp,q (s, a : F ),

a=1

(31) Tp,q (s, χ) = 2

p−1 X

χ(a)Tn,q (s, a : F ), where k, n ≥ 1 .

a=1

From (24)-(31), we derive   np ∞ X X r −r − 1 (−1)j =− (−1)n [pn]kq lp,q (r + k, w−r−k ) 2 r [j]q r+k k j=1 (j,p)=1

−

∞ X k=1

k=1

  r −r − 1 (−1)n [pn]kq Kp,q (r + k, w−r−k ) − Tp,q (r, w−r ). r+k k

Therefore we obtain the following theorem: Theorem 5. Let p be an odd prime and let n ≥ 1 be positive even integer. Then we have   np ∞ X X r −r − 1 (−1)j =− (−1)n [pn]kq lp,q (r + k, w−r−k ) 2 [j]rq r+k k j=1 (j,p)=1

(32) −

∞ X k=1

k=1

  r −r − 1 (−1)n [pn]kq Kp,q (r + k, w−r−k ) − Tp,q (r, w−r ), r+k k

where r is positive integer. Remark. When r is non-positive integer, we can easily derive the value of the left side of Eq.(32) from Eq.(13). For q = 1 in (32), we have   np ∞ X X (−1)j r −r − 1 2 (−1)n (pn)k lp (r + k, w−r−k ), =− jr k+r k j=1 (j,p)=1

k=1

where n is positive even integer (see [10]).

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L.C. JANG AND T. KIM

References [1]

M. Cenkci, Y. Simsek, V. Kurt, Further remarks on multiple p-adic q-L-function of two variables,, Advan. Stud. Contemp. Math. 14 (2007), 49-68.

[2]

T. Kim, S.-H. Rim, Y. Simsek, A note on the the alternating sums of powers of consecutive q-integers, Advan. Stud. Contemp. Math. 13 no.2 (2006), 340-348.

[3]

T. Kim, Power series and asymptotic series associated with the q-analogue of the two variable p-adic L-function, Russian J. Math. Phys. 12 (2005), 189-196.

[4]

T. Kim, q-Volkenborn Integration, Russ. J. Math. Phys. 9 (2005), 288-299.

[5]

T. Kim, On a q-analogue of the p-adic log gamma functions, J. Number Theory 76 (1999), 320-329.

[6]

T.Kim, Sums powers of consecutive q-integers, Advan. Stud. Contemp. Math. 9 (2004), 15-18.

[7]

T. Kim, On p-adic q-L-functions and sums of powers, Discrete Math. 252 (2002), 179-187.

[8]

T. Kim, Multiple p-adic L-functions, Russian J. Math. Phys. 13 (2) (2006).

[9]

T. Kim, A note on q-Volkenborn integration, Proc. Jangjeon Math. Soc. 8 (2005), 13-17.

[10]

S.-H. Rim, T. Kim, Explicit p-adic expansion for the alternating sums of powers, Advan. Stud. Contemp. Math. 14 (2007), 241-250.

[11]

Y. Simsek, On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers, Russian J. Math. Phys. 13 (2006), 340-348.

[12]

T. Kim, J. Y. Choi, J. Y. Sug, Extended q-Euler numbers and polynomials associated with fermionic p-adic q-integral on Zp , Russian J. Math. Phys. 14 (2007), 160-163.

[13]

L. C. Washington, p-adic L-functions and sums of powers, J. Number Theory 69 (1998), 50-61.

[14]

L. C. Washington, Introduction to cyclotomic fields, Springer-Verlag(1’st Ed.), 1982.

[15]

M. Schork, Ward’s “calculus of sequences”, q-calculus and the limit q → −1, Advan. Stud. Contemp. Math. 13 (2006), 131–141.

[16]

Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers, Advan. Stud. Contemp. Math. 11 (2005), 314-321.

[17]

J.Y. Sug, S. D. Choi,, Quantum yransport theory based on the equilibrium density projection technique, Physical Review E 55 (1997), 205-218.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.3,367-375,2008, COPYRIGHT 2008 EUDOXUS PRESS, 367 LLC

Fuzzy multi-metric spaces

Hakan Efe1 , Cihangir Alaca2 , Cemil Yildiz1 Department of Mathematics, Faculty of Science and Arts, Gazi University, Teknikokullar, 06500 Ankara, Turkey. [email protected], [email protected] 2 Department of Mathematics, Faculty of Science and Arts, Ondokuz Mayis University, Kurupelit, 55139 Samsun, Turkey. [email protected] 1

Abstract The purpose of this paper is to introduce the concept of fuzzy multimetric space by combining Smarandache multi-spaces with fuzzy metric space. Also some characteristics of fuzzy multi-metric space are obtained. Furthermore, we extend the Banach fixed point theorem to (fuzzy) contractive mappings in fuzzy multi-metric spaces. Keywords. Multi space; multi-metric space; triangular norm; fuzzy metric space. M.S.C. (2000). 54A40; 54E35; 54E40; 54E45. 1. Introduction In 1965, the concept of fuzzy set was introduced by Zadeh [13]. Many authors have introduced the concept of fuzzy metric space in different ways [1-3,5-8]. George and Veeramani [3,4] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [8] and defined a Hausdorff topology on this fuzzy metric space. They also showed that every metric induces a fuzzy metric. The notion of multi-spaces is introduced by Smarandache in [10] under his idea of hybrid mathematics: combining different fields into a unifying field [11]. The definition of multi-metric space is given by Mao [9], combining Smarandache multi-spaces with the classical metric spaces. He also give some characteristics of a multi-metric space. In this paper, we give the notion of fuzzy multi-metric space by combining Smarandache multi-spaces with the fuzzy metric space in the sense of George and Veeremani [3]. We give some theorems on covergence and continuity in fuzzy multi-metric space. Furthermore, we extend the Banach fixed point theorem to (fuzzy) contractive mappings, in our sense and Grabiec’s sense [5], on complete fuzzy multi-metric spaces (in George and Veeramani’s sense). 2. Preliminaries Definition 1 ([12]). A binary operation ∗ : [0, 1] × [0, 1] −→ [0, 1] is continuous t-norm if ∗ is satisfying the following conditions: (i) ∗ is commutative and associative; (ii) ∗ is continuous;

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(iii) a ∗ 1 = a for all a ∈ [0, 1]; (iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, and a, b, c, d ∈ [0, 1]. Definition 2 ([12]). A 3-tuple (X, M, ∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × (0, ∞) satisfying the following conditions: for all x, y, z ∈ X, s, t > 0, (i) (ii) (iii) (iv) (v)

M(x, y, t) > 0, M(x, y, t) = 1 if and only if x = y, M(x, y, t) = M(y, x, t), M(x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s), M(x, y, .) : (0, ∞) → [0, 1] is continuous.

Remark 1. In fuzzy metric space X, M(x, y, .) is non-decreasing for all x, y ∈ X. Example 1. Let (X, d) be a metric space. Denote a ∗ b = ab for all a, b ∈ [0, 1] and let Md be a fuzzy set on X 2 × (0, ∞) defined as follows: Md (x, y, t) =

ktn ktn + md(x, y)

for all k, m, n ∈ R+ . Then (X, Md , ∗) is a fuzzy metric space. Remark 2. Note the above example holds even with the t-norm a ∗ b = min{a, b} and hence M is a fuzzy metric with respect to any continuous t-norm. In the above example by taking k = m = n = 1, we get t Md (x, y, t) = t + d(x, y) We call this fuzzy metric induced by a metric d the standard fuzzy metric. Definition 3 ([3]). Let (X, M, ∗) be a fuzzy metric space and let r ∈ (0, 1), t > 0 and x ∈ X. The set B(x, r, t) = {y ∈ X : M(x, y, t) > 1 − r} is called the open ball with center x and radius r with respect to t. Theorem 1 ([3]). Every open ball B(x, r, t) is an open set. Remark 3. Let (X, M, ∗) be a fuzzy metric space. Define τ = {A ⊂ X : for each x ∈ X, there exist t > 0, r ∈ (0, 1) such that B(x, r, t) ⊂ A}. Then τ is a topology on X. Remark 4. (i) Since {B(x, n1 , n1 ) : n = 1, 2, ...} is a local base at x, the topology τ is first countable. (ii) Every fuzzy metric space is Hausdorff.

FUZZY MULTI-METRIC SPACES

(iii) Let (X, M, ∗) be an fuzzy metric space and τ be the topology on X induced by the fuzzy metric. Then for a sequence (xn )n in X, xn −→ x if and only if M(xn , x, t) −→ 1 as n −→ ∞. (iv) In a fuzzy metric space every compact set is closed and bounded. Definition 4 ([3]). Let (X, M, N, ∗, ♦) be a fuzzy metric space. Then,

(i) A sequence (xn )n in X is said to be Cauchy if for each ε > 0 and each t > 0, there exist n0 ∈ N such that M(xn , xm , t) > 1 − ε for all n, m ≥ n0 . (ii) (X, M, ∗) is called complete if every Cauchy sequence convergent with respect to M. e = ∪m Definition 5 ([10]). A multi-metric space is a union X i=1 Xi such that each Xi is a metric space with metric di for all i, 1 ≤ i ≤ m. When e = ∪m we say a multi-metric space X i=1 Xi , it means that a multi-metric space with metrics d1 , d2 , ..., dm such that (Xi , di ) is a metric space for any integer i, 1 ≤ i ≤ m. 3. Main results e = ∪m Definition 6. A fuzzy multi-metric space is a union X i=1 Xi such that each Xi is a fuzzy metric space with fuzzy metric Mi and t-norm ∗, for all i, 1 ≤ i ≤ m. e = ∪m When we say a fuzzy multi-metric space X i=1 Xi , it means that a fuzzy multi-metric space with fuzzy metrics M1 , M2 , ..., Mm such that (Xi , Mi , ∗) is a fuzzy metric space for any integer i, 1 ≤ i ≤ m.

Remark 5. The following two extremal cases are permitted in Definition 6: (i) There are integers i1 , i2 , ..., is such that Xi1 = Xi2 = · · · = Xis , where ij ∈ {1, 2, ..., m}, 1 ≤ j ≤ s. (ii) There are integers l1 , l2 , ..., ls such that Ml1 = Ml2 = · · · = Mls , where lj ∈ {1, 2, ..., m}, 1 ≤ j ≤ s.

e = ∪m Definition 7. Let X i=1 Xi be a fuzzy multi-metric space. We e and radius r, r ∈ (0, 1), define open ball B(x, r, t) with centre x ∈ X e : there exists an integer k, 1 ≤ k ≤ m t > 0 as B(x, r, t) = {y ∈ X such that Mk (x, y, t) > 1 − r}. e = ∪m Remark 6. Let X i=1 Xi be a fuzzy multi-metric space. Define e e there exist t > 0, r ∈ (0, 1) such that τ = {A ⊂ X : for each x ∈ X, e B(x, r, t) ⊂ A}. Then τ is a topology on X.

e = ∪m Remark 7. Let X i=1 Xi be a multi-metric space such that each (Xi , di ) is a metric space for all i, 1 ≤ i ≤ m. We define a ∗ b = ab and t Mi (x, y, t) = t + di (x, y)

369

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EFE ET AL

e is a fuzzy multi-metric space. We call those Mi as the standard then X e fuzzy multi-metric induced by di . Even if we take a ∗ b = min(a, b), X will be a fuzzy multi-metric space. The ´ (Xi , di ) is complete ³ metric space iff the standard fuzzy metric space Xi , Midi , ∗ is complete for all i, e is complete if and only if 1 ≤ i ≤ m. Then the multi-metric space X the standard fuzzy multi-metric space is complete. Corollary 1. If M1 , M2 , ..., Mm are m fuzzy metrics on a space X, then M1 ∗ M2 ∗ ... ∗ Mm is a fuzzy metric on X, where ∗ is min or product t-norm.

Corollary 2. If d1 , d2 , ..., dm are m metrics on a space X, then t t t ∗ ∗ ··· ∗ t + d1 (x, y) t + d2 (x, y) t + dm (x, y) is a fuzzy metric on X where a ∗ b = ab.

e = ∪m Definition 8. Let X i=1 Xi be a fuzzy multi-metric space and (xn )n e e if for be a sequence in X. (xn )n is said converge to a point x, x ∈ X any ε, ε ∈ (0, 1) there exist numbers n0 and i, 1 ≤ i ≤ m such that if n ≥ n0 then Mi (xn , x, t) > 1 − ε, for each t > 0. e we denote it by limn xn = For (xn )n convergence to a point x, x ∈ X, x or xn −→ x.

e = ∪m Theorem 2. X i=1 Xi be a fuzzy multi-metric space and (xn )n be e a sequence in X. Then xn −→ x iff there exist integers i, 1 ≤ i ≤ m such that Mi (xn , x, t) −→ 1 as n −→ ∞.

e = Theorem 3. A sequence (xn )n in a fuzzy multi-metric space X m ∪i=1 Xi is convergent if and only if there exist integers n0 and k, 1 ≤ k ≤ m, such that the subsequence {xn : n ≥ n0 } is a convergent sequence in (Xk , Mk , ∗).

Proof. If (xn )n is a convergent sequence in the fuzzy multi-metric space e by definition for any ε, ε ∈ (0, 1), there exist a point x, x ∈ X e and X, natural numbers n0 (ε) and k, 1 ≤ k ≤ m, such that if n ≥ n0 (ε), then Mk (xn , x, t) > 1 − ε for all t > 0. That is, {xn : n ≥ n0 (ε)} ⊂ Xk and {xn : n ≥ n0 (ε)} is a convergent sequence in (Xk , Mk , ∗). If there exist integer n0 and k, 1 ≤ k ≤ m, such that {xn : n ≥ n0 } is a convergent subsequence in (Xk , Mk , ∗), then for any ε, ε ∈ (0, 1), e by definition there exists a positive integer p0 and a point x, x ∈ X such that Mk (xn , x, t) > 1 − ε, for all t > 0, where n ≥ max{n0 , p0 }. Hence, (xn )n is a convergent sequence in the fuzzy multi-metric space e X. ¤ e = ∪m Theorem 4. Let X i=1 Xi be a fuzzy multi-metric space, (xn )n , e (yn )n are sequences in X and (tn )n ⊂ (0, ∞). If xn −→ x0 , yn −→ y0 ,

FUZZY MULTI-METRIC SPACES

371

tn −→ t and there is an integer p, 1 ≤ p ≤ m such that x0 , y0 ∈ Xp , t > 0, then limn Mp (xn , yn , tn ) = Mp (x0 , y0 , t). Proof. Since xn −→ x0 and yn −→ y0 there exist integers n1 and n2 such that if n ≥ max{n1 , n2 }, then xn , yn ∈ Xp . Fix δ > 0 such that δ < t/2. Then, there exists an integer n3 such that |tn − t| < δ for all n ≥ n0 = max{n1 , n2 , n3 }. Hence, Mp (xn , yn , tn ) ≥ Mp (xn , x0 , δ) ∗ Mp (x0 , y0 , t − 2δ) ∗ Mp (yn , y0 , δ)

and

Mp (x0 , y0 , t + 2δ) ≥ Mp (xn , x0 , δ) ∗ Mp (xn , yn , tn ) ∗ Mp (yn , y0 , δ)

for all n ≥ n0 . By taking limits when n −→ ∞, we get

lim Mp (xn , yn , tn ) ≥ 1 ∗ Mp (x0 , y0 , t − 2δ) ∗ 1 n

and Mp (x0 , y0 , t + 2δ) ≥ 1 ∗ lim Mp (xn , yn , tn ) ∗ 1 n

respectively. So, by continuity of the function t −→ Mp (x, y, t) we obtain lim Mp (xn , yn , tn ) = Mp (x0 , y0 , t). n

¤

Theorem 5. If (xn )n is a convergent sequence in a fuzzy multi metric e = ∪m space X i=1 Xi , then (xn )n has only one limit point.

e Then there exist Proof. Let xn −→ x1 , xn −→ x2 and x1 , x2 ∈ X. integer n0 and i, 1 ≤ i ≤ m such that xn ∈ Xi for all n ≥ n0 . Hence, ¶ ¶ µ µ t t ∗ Mi xn , x2 , . 1 ≥ Mi (x1 , x2 , t) ≥ Mi x1 , xn , 2 2 By taking limits when n −→ ∞, we get 1 ≥ Mi (x1 , x2 , t) ≥ 1 ∗ 1 = 1

which implies x1 = x2 .

¤

Theorem 6. Any convergent sequence in a fuzzy multi metric space is a bounded points set. Proof. It is clear from Theorem 5.

¤

e = Definition 9. A sequence (xn )n in a fuzzy multi-metric space X m ∪i=1 Xi is called Cauchy sequence if for each ε, ε ∈ (0, 1), t > 0, there exist integers n0 (ε) and s, 1 ≤ s ≤ m such that Ms (xn , xm , t) > 1 − ε for all n, m ≥ n0 (ε). Theorem 7. A Cauchy sequence (xn )n in a fuzzy multi-metric space e = ∪m X i=1 Xi is convergent if and only if for all k, 1 ≤ k ≤ m, |(xn )n ∩ Xk | is finite or infinite but (xn )n ∩ Xk is convergent in (Xk , Mk , ∗).

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Proof. The necessity of conditions is clear from Theorem 3. Now, we prove the sufficiency. By definition, there exist integers s, 1 ≤ s ≤ m and n1 such that xn ∈ Xs for n ≥ n1 . If |(xn )n ∩ Xk | is infinite and limn→∞ ((xn )n ∩ Xk ) = x, then there must be k = s. Denoted by (xn )n ∩ Xk = {xk1 , xk2 , ..., xkn , ...}. For any δ, δ ∈ (0, 1), there exists an integer n2 , n2 ≥ n1 such that Mk (xm , xn , 2t ) > 1 − δ and Mk (xkn , x, 2t ) > 1 − δ for all m, n ≥ n2 and for all t > 0. Since δ ∈ (0, 1), we can find a ε, ε ∈ (0, 1), such that (1 − δ) ∗ (1 − δ) > 1 − ε. Then, by using Theorem 4, we get that ¶ ¶ µ µ t t ∗ Mk xkn , x, Mk (xn , x, t) ≥ Mk xn , xkn , 2 2 ≥ (1 − δ) ∗ (1 − δ) > 1 − ε. Hence, limn xn = x which completes the proof.

¤

Definition 10. A fuzzy multi-metric space is said to be complete if every Cauchy sequence is convergent. e = ∪m Theorem 8. Let X i=1 Xi be a complete fuzzy multi-metric space. For a ball sequence (B(xn , εn , t))n , where 0 < εn < 1 for n = 1, 2, ..., the following conditions hold: (i) B(x1 , ε1 , t) ⊃ B(x2 , ε2 , t) ⊃ ... ⊃ B(xn , εn , t) ⊃ ... (ii) limn εn = 0. Then, ∩∞ n=1 B(xn , εn , t) only has one point. Proof. First, we prove that the sequence (xn )n is a Cauchy sequence e By the condition (i), we know that if m ≥ n, then xm ∈ in X. B(xm , εm , t) ⊂ B(xn , εn , t) for all t > 0. Whence, for all i, 1 ≤ i ≤ m, Mi (xm , xn , t) > 1 − εn for xm , xn ∈ Xi . For any ε, ε ∈ (0, 1), since limn εn = 0, there exists an integer n0 (ε) such that if n > n0 (ε), then εn < ε. Therefore, if xn ∈ Xl , then limm xm = xn . Whence, there exists an integer n0 such that m ≥ n0 , xm ∈ Ml by Theorem 3. Take integers m, n ≥ max{n0 , n0 (ε)}. We know that Ml (xm , xn , t) > 1 − εn > 1 − ε. So, (xn )n is a Cauchy sequence. e is complete. We know that the sequence (xn )n By the assumption, X e By conditions (i) and (ii), we have is convergence to a point x0 , x0 ∈ X. that Ml (x0 , xn , t) > 1 − εn as m −→ ∞. Hence, x0 ∈ ∩∞ n=1 B(xn , εn , t). Now if there is a point y ∈ ∩∞ B(x , ε , t), then there must be n n n=1 y ∈ Xl . We get that 1 ≥ Ml (y, x0 , t) = lim Ml (y, xn , t) ≥ lim(1 − εn ) = 1 n

n

for all t > 0, by Theorem 4. Therefore, Ml (y, x0 , t) = 1 which implies y = x0 . ¤

FUZZY MULTI-METRIC SPACES

f1 and X f2 be two fuzzy multi-metric spaces and f Definition 11. Let X f2 , x0 ∈ X f1 and f(x0 ) = y0 . For ε, ε ∈ (0, 1) f1 to X be a mapping from X if there exists a number δ, δ ∈ (0, 1) such that for all x ∈ B(x0 , δ, t), f2 , i.e., f (B(x0 , δ, t)) ⊂ B(y0 , ε, t), for all f (x) = y ∈ B(y0 , ε, t) ⊂ X t > 0, then we say f is continuous at point x0 . If f is continuous at f1 to f1 , then f is said to a continuous mapping from X every point of X f X2 . f1 and X f2 be two fuzzy multi-metric spaces, f be Proposition 1. Let X f2 and (xn )n be a sequence in X f1 . f1 to X a continuous mapping from X If xn −→ x, then f (xn ) −→ f (x).

4. Fixed points for fuzzy multi-metric spaces e = ∪m Definition 12. Let X i=1 Xi be a fuzzy multi-metric space and e e e is called a fixed point of T if T : X → X be a mapping. x∗ ∈ X e by T x∗ = x∗ . Denote the number of fixed points of a mapping T in X ‡ Φ (T ).

e = ∪m Definition 13. Let X i=1 Xi be a fuzzy multi-metric space. We e e is fuzzy contractive if there exist will say the mapping f : X −→ X k ∈ (0, 1), 1 ≤ i, j ≤ m, such that µ ¶ 1 1 −1≤k −1 Mj (f (x), f (y), t) Mi (x, y, t) e and t > 0. k ∈ (0, 1), is called contractive constant for each x, y ∈ X of f .

e = ∪m Proposition 2. Let X i=1 Xi be a multi-metric space where each (Xi , di ) is a metric space for any integer i, 1 ≤ i ≤ m. The mapping e −→ X e is contractive (a contraction) on the multi-metric space X e f :X with contractive constant k iff f is fuzzy contractive, with contractive m e constant ´ standard fuzzy multi-metric space X = ∪i=1 Xi such ³ k, on the that Xi , Midi , ∗ standard fuzzy metric space induced by di for all 1 ≤ i ≤ m. e = ∪m Definition 14. A sequence (xn )n in a multi-metric space X i=1 Xi is said to be contractive if there exists k ∈ (0, 1) such that dj (xn+1 , xn+2 ) ≤ kdi (xn , xn+1 ), for all n ∈ N and 1 ≤ i, j ≤ m.

e = ∪m Definition 15. Let X i=1 Xi be a fuzzy multi-metric space. We e is fuzzy contractive if there exists will say that the sequence (xn )n in X k ∈ (0, 1) such that µ ¶ 1 1 −1≤k −1 Mj (xn+1 , xn+2 , t) Mi (xn , xn+1 , t) for all t > 0, n ∈ N and 1 ≤ i, j ≤ m.

373

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EFE ET AL

e = ∪m Proposition 3. Let X i=1 Xi be the standard fuzzy multi-metric space induced by the metric di on Xi for all 1 ≤ i ≤ m. The sequence e is contractive in multi-metric space iff (xn )n is fuzzy con(xn )n in X tractive in standard fuzzy multi-metric space.

Next, we extend the Banach fixed point theorem to fuzzy contractive mappings of complete fuzzy multi-metric spaces.

e = ∪m Theorem 9. Let X i=1 Xi be a complete fuzzy multi-metric space e −→ X e in which fuzzy contractive sequences are Cauchy. Let T : X be a fuzzy contractive mapping being k the contractive constant. Then 1 ≤ Φ‡ (T ) ≤ m. e Let xn = T n (x), n ∈ N. We have for t > 0 Proof. Fix x ∈ X. µ ¶ 1 1 −1≤k −1 Mi (T (x), T 2 (x), t) Mi (x, x1 , t) and by induction

1 −1≤k Mi (xn+1 , xn+2 , t)

µ

¶ 1 −1 , Mi (xn , xn+1 , t)

n ∈ N. Then (xn )n is a fuzzy contractive sequence, so it is a Cauchy sequence e We will see z ∗ is and, hence, (xn )n converges to z ∗ , for some z ∗ ∈ X. a fixed point for T . By Theorem 2, we have µ ¶ 1 1 −1≤k − 1 −→ 0 Mi (T (y), T (xn ), t) Mi (y, xn , t)

as n −→ ∞. Then limn Mi (T (y), T (xn ), t) = 1 for each t > 0, and, therefore, limn T (xn ) = T (z ∗ ), i.e., limn xn+1 = T (z ∗ ) and then T (z ∗ ) = z∗. For other chosen points u0 , v0 ∈ X1 , we can also define recursively un+1 = T un , vn+1 = T vn and get the limit points, there exists an integer i0 such that, limn un = limn vn = w∗ ∈ Xi0 , T u∗ ∈ Xi0 . Then for t > 0 we have 1 1 −1 = −1 ∗ ∗ ∗ Mi0 (z , u , t) Mi0 (T (z ), T (u∗ ), t) µ ¶ 1 ≤ k −1 Mi0 (z ∗ , u∗ , t) µ ¶ 1 = k −1 Mi0 (T (z ∗ ), T (u∗ ), t) ¶ µ 1 2 ≤ k −1 Mi0 (z ∗ , u∗ , t) ¶ µ 1 n ≤ ··· ≤ k − 1 −→ 0 Mi0 (z ∗ , u∗ , t)

FUZZY MULTI-METRIC SPACES

as n −→ ∞. Hence, Mi0 (z ∗ , u∗ , t) = 1 and then z ∗ = u∗ . Similar consider the points in Xi , 2 ≤ i ≤ m, we get 1 ≤ Φ‡ (T ) ≤ m. ¤ ³ ´ e Mi , ∗ is a complete standard fuzzy multi-metric Now suppose X, di

space where (Xi , Midi , ∗) fuzzy metric space induced by the metric di in Xi for all 1 ≤ i, j ≤ m. From Remark 7 (Xi , di ) is complete, then if (xn )n is a fuzzy contractive sequence, by Proposition 3 it is contractive in (Xi , di ), hence convergent. So, from Theorem 9 we have the following corollary, which can be considered the fuzzy version of the classic Banach contraction theorem on complete metric spaces.

e be a complete standard fuzzy multi-metric space Corollary 3. Let X e e and let T : X −→ X a fuzzy contractive mapping. Then 1 ≤ Φ‡ (T ) ≤ m. References

[1] Z. K. Deng, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl. 86(1982) 74−95. [2] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69(1979) 205 − 230. [3] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64(1994) 395 − 399. [4] A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90(1997) 365 − 368. [5] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27(1988) 385 − 389. [6] V. Gregori, A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125(2002) 245 − 252. [7] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12(1984) 225 − 229. [8] O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11(1975) 326 − 334. [9] L. Mao, On multi-metric spaces, eprint arXiv: math.GM/0510480, 1/2005. [10] F. Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000. [11] F. Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Probability, Set, and Logic, American research Press, Rehoboth 1999. [12] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math. 10(1960) 314 − 334. [13] L. A. Zadeh, Fuzzy sets, Inform and Control 8(1965) 338 − 353.

375

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.3,377-389,2008, COPYRIGHT 2008 EUDOXUS PRESS, 377 LLC

Statistical Convergence of Double Sequences on Intuitionistic Fuzzy Normed Spaces Sevda Karakus and Kamil Demirci Department of Mathematics, Faculty of Sciences and Arts Sinop, Ondokuz Mayis University, 57000 Sinop-TURKEY E-mail: [email protected] , [email protected] Abstract The concept of statistical convergence was presented by Steinhaus (1951). This concept was extended to the double sequences by Mursaleen and Edely (2003). In this paper, we de…ne and study statistical analogue of convergence and Cauchy for double sequences on intuitionistic fuzzy normed spaces. Then we give a useful characterization for statistically convergent double sequences. Furthermore, we display an example such that our method of convergence is stronger than the usual convergence for double sequences on intuitionistic fuzzy normed spaces. KEY WORDS: Natural double density, statistical convergence, continuous t norm, continuous t conorm, intuitionistic fuzzy normed space.

1

Introduction

In 1965, the concept of fuzzy sets was introduced by Zadeh [29]. Then many authors developed the theory of fuzzy set and applications. The fuzzy logic has been used many …elds, like metric and topological spaces [9] ; [10] ; [16] ; [19], theory of functions [4] ; [18] ; [28] ; computer programing [17], econometrics and other …elds [1] ; [2] ; [3] ; [8] ; [20] ; [22] : Also, recently, the concepts of intuitionistic fuzzy metric space has been studied by Park [23], and intuitionistic fuzzy normed space have been studied by Saadati and Park [25] : In this paper we give statistical analogues of convergence and Cauchy for double sequences which studied in Mursaleen and Osama [21] on intuitionistic fuzzy normed spaces:Also we display an example such that our method of convergence is stronger than the usual convergence for double sequences on intuitionistic fuzzy normed spaces. Now we recall some notations and de…nitions which we used in the paper. De…nition 1 [26]A binary operation : [0; 1] [0; 1] ! [0; 1] is said to be a continuous t norm if it satis…es the following conditions:

1

378

KARAKUS-DEMIRCI

(a)

is associative and commutative,

(b)

is continuous,

(c) a 1 = a for all a 2 [0; 1]; (d) a b

c d whenever a

c and b

d for each a; b; c; d 2 [0; 1]:

Two typical examples of continuous t norm are a min (a; b) for all a; b 2 [0; 1].

b = ab and a

b =

De…nition 2 [26]A binary operation : [0; 1] [0; 1] ! [0; 1] is said to be a continuous t conorm if it satis…es the following conditions: (a)

is associative and commutative,

(b)

is continuous,

(c) a 0 = a for all a 2 [0; 1]; (d) a b

c d whenever a

c and b

d for each a; b; c; d 2 [0; 1]:

Two typical examples of continuous t conorm are a b = min (a + b; 1) and a b = max (a; b) for all a; b 2 [0; 1]. Now we give the concept of intuitionistic fuzzy normed space which has recently introduced by Saadati and Park [25]. De…nition 3 [25] The 5 tuple (V; ; ; ; ) is said to be an intuitionistic fuzzy normed space (IFNS) if V is a vector space, is a continuous t norm, is a continuous t conorm, and ; fuzzy sets on V (0; 1) satisfying the following conditions for every x; y 2 V and s; t > 0 : (a)

(x; t) + (x; t)

(b)

(x; t) > 0;

(c)

(x; t) = 1 if and only if x = 0;

(d)

( x; t) =

(e)

(x; t)

(f )

(x; ) : (0; 1) ! [0; 1] is continuous,

(g) lim

t!1

1;

x; j t j

(y; s)

for each

6= 0,

(x + y; t + s);

(x; t) = 1 and lim (x; t) = 0; t!0

(h)

(x; t) < 1;

(i)

(x; t) = 0 if and only if x = 0;

(j)

( x; t) =

x; j t j

for each

6= 0, 2

STATISTICAL CONVERGENCE...

(k)

(x; t)

(y; s)

(x + y; t + s);

(l)

(x; ) : (0; 1) ! [0; 1] is continuous,

(m) lim (x; t) = 0 and lim (x; t) = 0: t!1

t!0

In this case ( ; ) is called an intuitionistic fuzzy norm. We can give an example as follow: Let (V; k k) be a normed space, and let a b = ab and a b = minfa + b; 1g for all a; b 2 [0; 1]: If we de…ne 0 (x; t)

:=

t and t + kxk

0 (x; t)

:=

kxk : t + kxk

for all x 2 V and every t > 0; then observe that (V; ; ; ; ) is an intuitionistic fuzzy normed space. Before we present the new de…nitions and the main theorems, we shall recall some concepts which we need. By the convergence of a double sequence we mean the convergence in Pring1 sheim’s sense [24]. A double sequence x = (xjk )jk=0 is called convergent in the Pringsheim’s sense if for every " > 0 there exists N 2 N such that jxjk Lj < " whenever j; k N . L is called the Pringsheim limit of x. A double sequence x = (xjk ) is said to be Cauchy sequence if for every " > 0 there exists N 2 N such that jxpq xjk j < " for all p j N , q k N . A double sequence x is bounded if there exists a positive number M such that jxjk j < M for all j and k. So we can give the ( ; ) analogue of above two de…nitions as follow: De…nition 4 Let (V; ; ; ; ) be an IFNS. Then, a double sequence x = (xjk ) is said to be convergent to L 2 V with respect to the intuitionistic fuzzy norm ( ; ) if, for every " > 0 and t > 0; there exists N 2 N such that (xjk L; t) > 1 " and (xjk L; t) < " for all j; k N: It is denoted by ( ; v)2 lim x = L ( ; )2

or xjk ! L as j; k ! 1: De…nition 5 Let (V; ; ; ; ) be an IFNS. Then, a double sequence x = (xjk ) is said to be a Cauchy sequence with respect to the intuitionistic fuzzy norm ( ; ) provided that, for every " > 0 and t > 0; there exists N = N (") and M = M (") such that (xjk xpq ; t) > 1 " and (xjk xpq ; t) < " for all j; p N , k; q M Now we …rst recall statistical convergence and then in new section, we introduce basic de…nitions and properties which we mention above .

2

Statistical Convergence of Double Sequence on IFNS

Steinhaus [27] introduced the idea of statistical convergence (see also Fast [11]). If K is a subset of N, the set of natural numbers, then the asymptotic density 3

379

380

KARAKUS-DEMIRCI

of K denoted by (K), is given by (K) := lim n

1 jfk n

n : k 2 Kgj

whenever the limit exists, when jAj denotes the cardinality of the set A. A sequence x = (xk ) of numbers is statistically convergent to L if (fk 2 N : jxk

Lj

"g) = 0

for every " > 0. In this case we write st lim x = L. Statistical convergence has been investigated in a number of paper [6] ; [7] ; [12] ; [13] ; [14] ; [15]. Now we recall the concept of statistical convergence of double sequences. Let K N N be a two-dimensional set of positive integers and let K (n; m) be the numbers of (i; j) in K such that i n and j m. Then the twodimensional analogue of natural density can be de…ned as follows. The lower asymptotic density of a set K N N is de…ned as 2

K (n; m) : nm

(K) = lim inf n;m

In case the sequence (K (n; m) nm) has a limit in Pringsheim’s sense [24] then we say that K has a double natural density and is de…ned as lim

n;m

K (n; m) = nm

2

(K) :

i2 ; j 2 : i; j 2 N , then p p K (n; m) n m (K) = lim lim = 0: 2 n;m n;m nm nm

If we consider the set of K =

Also, if we consider the set of f(i; 2j) : i; j 2 Ng has double natural density 1=2. If we set n = m, we have a two-dimensional natural density considered by Christopher [5]. Now we recall the concepts of statistically convergent and statistically Cauchy for double sequence as follows: De…nition 6 [21] A real double sequence x = (xjk ) is said to be statistically convergent the number ` provided that, for each " > 0, the set f(j; k) ; j

n and k

m : jxjk

`j

has double natural density zero. In this case we write st2

"g limxjk = `. j;k

De…nition 7 [21]A real double sequence x = (xjk ) is said to be statistically Cauchy provided that, for every " > 0 there exist N = N (") and M = M (") such that for all j; p N; k; q M; the set f(j; k) ; j

n and k

m : jxjk

has double natural density zero. 4

xpq j

"g

STATISTICAL CONVERGENCE...

381

Now we give the analogues of these with respect to the intuitionistic fuzzy norm ( ; ). De…nition 8 Let (V; ; ; ; ) be an IFNS. A real double sequence x = (xjk ) is statistically convergent to L 2 V with respect to the intuitionistic fuzzy norm ( ; ) provided that, for every " > 0 and t > 0; K = f(j; k) ; j

n and k

m : (xjk

L; t)

1

" or (xjk

L; t)

"g (1)

has double natural density zero, i.e., if K (n; m) be the numbers of (j; k) in K lim

n;m

In this case we write st(

; )2

K (n; m) = 0: nm

(2)

limxjk = L; where L is said to be st( j;k

; )2

limit.

Also we denote the set of all statistically convergent double sequences with respect to the intuitionistic fuzzy norm ( ; ) by st( ; )2 . By using (2) and the well-known properties of the double natural density, we easily get the following lemma. Lemma 9 Let (V; ; ; ; ) be an IFNS. Then, for every " > 0 and t > 0; the following statements are equivalent: (i) st( (ii)

; )2

limxjk = L j;k

n and k m : (xjk 2 f(j; k) ; j n and k m : (xjk L; t) "g = 0:

(iii)

2 f(j; k) ; j "g = 1:

(iv)

n and k m : (xjk 2 f(j; k) ; j n and k m : (xjk L; t) < "g = 1:

(v) st2

lim (xjk

n and k

m : (xjk

L; t) = 1 and st2

L; t)

1

L; t) > 1

"g =

" and (xjk

L; t) > 1

lim (xjk

2 f(j; k) ;

"g =

j

L; t)
0 and " > 0: Choose r 2 (0; 1) such that (1 r) (1 r) 1 " and r r ": Then, we de…ne the following sets: K K K K

;1 (r; t)

: = f(j; k) 2 N ;2 (r; t) : = f(j; k) 2 N ;1 (r; t) : = f(j; k) 2 N ;2 (r; t) : = f(j; k) 2 N

N: N: N: N: 5

(xjk (xjk (xjk (xjk

L1 ; t) L2 ; t) L1 ; t) L2 ; t)

1 rg ; 1 rg ; rg ; rg :

382

KARAKUS-DEMIRCI

Since st(

; )2

lim x = L1 ; we have 2 fK ;1 ("; t)g

Furthermore, using st(

; )2

2 fK ;2 ("; t)g

=

2 fK ;1 ("; t)g

=0

for all t > 0:

lim x = L2 ; we get =

2 fK ;2 ("; t)g

=0

for all t > 0:

Now let K ; ("; t) := fK ;1 ("; t) [ K ;2 ("; t)g \ fK ;1 ("; t) [ K ;2 ("; t)g : Then observe that 2 fK ; ("; t)g = 0 which implies 2 fN N=K ; ("; t)g = 1: If (j; k) 2 N N=K ; ("; t); then we have two possible cases. The former is the case of (j; k) 2 N N=fK ;1 ("; t) [ K ;2 ("; t)g; and the letter is (j; k) 2 N N= fK ;1 ("; t) [ K ;2 ("; t)g : We …rst consider that (j; k) 2 N

N= fK

;1 ("; t)

[K

;2 ("; t)g :

Then we have (L1

L2 ; t)

t t (xjk L2 ; ) L1 ; ) 2 2 r) (1 r) 1 ":

(xjk > (1

Since " > 0 was arbitrary, we get (L1 L2 ; t) = 1 for all t > 0;which yields L1 = L2 : On the other hand, if (j; k) 2 N N=fK ;1 ("; t) [ K ;2 ("; t)g; then we may write that (L1

L2 ; t)

t L1 ; ) 2 ":

(xjk
0 and t > 0; there is a number N 2 N such that (xjk L; t) > 1 " and (xjk L; t) < " for all j

N and k f(j; k) 2 N

N: This guarantees that the set N : (xjk

L; t)

1

" or (xjk

L; t)

"g

has at most …nitely many terms. Since every …nite subset of the natural numbers has double density zero, we immediately see that 2 f(j; k)

2N

N : (xjk

L; t)

1

" or (xjk

L; t)

"g = 0;

whence the result. The following example shows that the converse of Theorem 11 is not hold in general. 6

STATISTICAL CONVERGENCE...

383

Example 12 Let (R,j j) denote the space of real numbers with the usual norm, and let a b = ab and a b = minfa + b; 1g for all a; b 2 [0; 1]: For all x 2 R and every t > 0; consider 0 (x; t)

:=

t and t + jxj

0 (x; t)

:=

jxj : t + jxj

In this case observe that (R; ; ; ; ) is an IFNS. Now de…ne a double sequence x = (xjk ) whose terms are given by xjk :=

1; if j and k are squares 0; otherwise.

(3)

Then, for every 0 < " < 1 and for any t > 0; let Kn ("; t) := f(j; k) ; j n and k m : 0 (xjk ; t) 1 " or 0 (xjk ; t) "g: Since Kn ("; t)

t t + jxjk j

jxjk j t + jxjk j

=

(j; k) ; j

n and k

m:

=

(j; k) ; j

n and k

= f(j; k) ; j = f(j; k) ; j

n and k n and k

>0 1 " m : xjk = 1g m : j and k are squaresg

we have 2 (Kn ("; t))

m : jxjk j

1

" or

"t

"

p p n m = 0: n;m nm lim

Hence, we get st( 0 ; 0 )2 lim x = 0: However, since the sequence x = (xjk ) given by (3) is not convergent in the space (R,j j); by Lemma 4.10 of [25], we also see that x is not convergent with respect to the intuitionistic fuzzy norm ( 0 ; 0 ): Theorem 13 Let (V; ; ; ; ) be an IFNS. Then st( ; )2 lim x = L if and only if there exists a subset K = f(j; k)g N N, j; k = 1; 2; :::; such that lim xjk = L. 2 (K) = 1 and ( ; )2 j;k!1 (j;k)2K

Proof. We …rst assume that st( let Kr :=

; )2

lim x = L: Now, for any t > 0 and j 2 N;

(j; k) 2 N

N : (xjk

L; t)

1

(j; k) 2 N

N : (xjk

L; t) > 1

1 or (xjk r

L; t)

1 r

and Mr

=

(r = 1; 2; :::) : Then

2

(Kr ) = 0 and 7

1 and (xjk r

L; t)
1

" and (xjk

L; t) < "g

and " > 1r (r = 1; 2; :::) : Then (3) 2 (M" ) = 0, and by (1), Mr M" . Hence 2 (Mr ) = 0 which contradicts (2). Therefore (xjk ) is convergent to L. Conversely, suppose that there exists a subset K = f(j; k)g N N such that 2 (K) = 1 and ( ; )2 limxjk = L, i.e. there exists N 2 N such j;k

that for every " > 0 and t > 0 (xjk

L; t) > 1

" and (xjk

L; t) < ";

8j; k

N:

Now K"

Therefore

= f(j; k) 2 N N : (xjk L; t) 1 " or (xjk N N f(jN +1 ; kN +1 ) ; (jN +2 ; kN +2 ) ; :::g : 2

(K" )

1

L; t)

"g

1 = 0. Hence x is statistically convergent to L:

De…nition 14 Let (V; ; ; ; ) be an IFNS. We say that a double sequence x = (xjk ) is statistically Cauchy with respect to the intuitionistic fuzzy norm ( ; ) provided that, for every " > 0 and t > 0; there exist N = N (") and M = M (") such that for all j; p N , k; q M , the set f(j; k) ; j

n; k

m:

(xjk

xpq ; t)

1

" or

(xjk

xpq ; t)

"g

has double natural density zero. Now using a similar technique in the proof of Theorem 13 one can get the following result at once. Theorem 15 Let (V; ; ; ; ) be an IFNS, and let x = (xjk ) be a double sequence whose terms are in the vector space V . Then, the following conditions are equivalent: (i) x is a statistically Cauchy sequence with respect to the intuitionistic fuzzy norm ( ; ). 8

STATISTICAL CONVERGENCE...

385

(ii) There exists an increasing index sequence K = f(j; k)g N N, j; k = 1; 2; ::: such that 2 (K) = 1 and the subsequence fxjk g(j;k)2K is a Cauchy sequence with respect to the intuitionistic fuzzy norm ( ; ): Now we show that statistically convergence of double sequences on IFNS has some arithmetical properties similar to properties of the usual convergence on R. Lemma 16 Let (V; ; ; ; ) be an IFNS. If st( ; )2 lim yjk = then st( ; )2 lim (xjk + yjk ) = + : Proof. Let st( ; )2 lim xjk = , st( ; Choose r 2 (0; 1) such that (1 r) (1 de…ne the following sets: K K K K Since st(

;1 (r; t)

: = f(j; k) 2 N (r; t) : = f(j; k) 2 N ;2 ;1 (r; t) : = f(j; k) 2 N ;2 (r; t) : = f(j; k) 2 N

; )2

)2

r)

lim xjk =

and st(

; )2

lim yjk = , t > 0 and " 2 (0; 1). 1 " and r r ". Then we

N: N: N: N:

(xjk (yjk (xjk (yjk

; t) ; t) ; t) ; t)

1 rg ; 1 rg ; rg ; rg :

lim xjk = ; we have 2 fK ;1 ("; t)g

Similarly, since st(

; )2

=

2 fK ;1 ("; t)g

=0

for all t > 0:

=0

for all t > 0:

lim yjk = ; we get

2 fK ;2 ("; t)g

=

2 fK ;2 ("; t)g

Now let K ; ("; t) := fK ;1 ("; t) [ K ;2 ("; t)g \ fK ;1 ("; t) [ K ;2 ("; t)g : Then observe that 2 fK ; ("; t)g = 0 which implies 2 fN N=K ; ("; t)g = 1: If (j; k) 2 N N=K ; ("; t); then we have two possible cases. The former is the case of (j; k) 2 N N=fK ;1 ("; t) [ K ;2 ("; t)g; and the letter is (j; k) 2 N N= fK ;1 ("; t) [ K ;2 ("; t)g : We …rst consider that (j; k) 2 N

N= fK

) + (yjk

) ; t)

;1 ("; t)

[K

;2 ("; t)g :

Then we have ((xjk

> (1 On the other hand, if (j; k) 2 N that ((xjk

) + (yjk

t (yjk ; ) 2 r) (1 r) 1

t ; ) 2

(xjk

N=fK

) ; t)

;1 ("; t) [ K ;2 ("; t)g;

t ; ) 2

(xjk < 9

r r

":

":

then we can write

(xjk

t ; ) 2

386

KARAKUS-DEMIRCI

This show that

so st(

2

(j; k) 2 N

N : ((xjk ) + (yjk ) ; t) 1 or ((xjk ) + (yjk ) ; t)

; )2

lim (xjk + yjk ) = + .

Lemma 17 Let (V; ; ; ; ) be an IFNS. If st( then st( ; )2 lim xjk = : Proof. Let st( ; )2 lim xjk = the case of = 0. In this case (0xjk

" "

lim xjk =

; )2

=0

and

2R

, " 2 (0; 1) and t > 0. First of all we consider

0 ; t) = (0; t) = 1 > 1

":

Similarly we observe that (0xjk for st(

0 ; t) = (0; t) = 0 < "

= 0: So we obtain ( ; )2 lim 0xjk = 0: Now we consider the case of

lim 0x = 0. Then from Theorem 11 we have

; )2

2

(f(j; k) 2 N

N:

2 R ( 6= 0) : From de…nition we can write

(xjk

; t)

1

"

or

(xjk

; t)

"g) = 0:

So, if we de…ne the sets: K K

;1 ("; t)

: = f(j; k) 2 N ;1 ("; t) : = f(j; k) 2 N

N : (xjk N : (xjk

; t) ; t)

1 "g "g

then we can say 2 fK ;1 ("; t)g = 2 fK ;1 ("; t)g = 0 for all t > 0. Now let K ; ("; t) = K ;1 ("; t) [ K ;1 ("; t) then 2 fK ; ("; t)g = 0 which implies N K ; ("; t)g = 1. If (j; k) 2 N N K ; ("; t) then for 2 R ( 6= 0) 2 fN ( xjk

; t)

=

= = Similarly, we observe that for ( xjk

; t)

=

= =

t ) j j

(xjk

;

(xjk

; t)

(xjk (xjk

; t) 1 ; t) > 1

(0;

t j j

t)

":

2 R ( 6= 0) t ) j j

(xjk

;

(xjk

; t)

(xjk (xjk

; t) 0 ; t) < ":

10

(0;

t j j

t)

STATISTICAL CONVERGENCE...

387

This show that 2

(f(j; k) 2 N

so st(

N : ( xjk

lim xjk =

; )2

; t)

1

"

or

( xjk

; t)

"g) = 0

.

Lemma 18 Let (V; ; ; ; ) be an IFNS. If st( ; )2 lim yjk = then st( ; )2 lim (xjk yjk ) = .

lim xjk =

and st(

; )2

Proof. The proof is clear from Lemma 16 and Lemma 17. De…nition 19 Let (V; ; ; ; ) be an IFNS . We say that a double sequence x = (xjk ) is IF-bounded if there exist t > 0 and 0 < r < 1 such that (xjk ; t) > 1 r and (xjk ; t) < r for every (j; k) 2 N N. De…nition 20 Let (V; ; ; ; ) be an IFNS . For t > 0, we de…ne open ball B (x; r; t) with center x 2 V and radius 0 < r < 1, as B (x; r; t) = fy 2 V :

(x

y; t) > 1

r;

(x

y; t) < rg :

It follows from Lemma 16 Lemma 17 and Lemma 18, that the set of all IF-bounded statistically convergent double sequences on IFNS is a linear sub( ; ) space of the linear normed space `1 2 (V ) of all IF-bounded sequences on IFNS. Theorem 21 Let (V; ; ; ; ) be an IFNS and the set st( is closed linear subspace of the set Proof. It is clear that st( Now we show that st( y 2 st(

; )2

(V ) \

( ; ) `1 2

; )2

; )2

( ; ) `1 2

(V ).

( ; )2

(V ) \ `1

(V ) \

( ; ) `1 2

; )2

(V )

(V )

st(

(V ) . Since B (y; r; t)\ st(

st( ; )2 ; )2

( ; ) `1 2

; )2

( ; )2

(V ) \ `1

( ; )2

(V ) \ `1

(V ) \ (V ) \

( ; ) `1 2

( ; ) `1 2

(V )

(V ) :

(V ). Let (V ) 6= ?

, there is a x 2 B (y; r; t) \ st( ; )2 (V ) \ (V ) . Let t > 0 and " 2 (0; 1). Choose r 2 (0; 1) such that (1 r) (1 r) 1 " ( ; ) and r r ". Since x 2 B (y; r; t) \ st( ; )2 (V ) \ `1 2 (V ) , there is a set K N N with (K) = 1 such that yjk

xjk ;

t 2

>1

r

and

yjk

xjk ;

t 2

>1

r

and

xjk ;

and

xjk ; t 2

t 2

(1 r) (1 r) 1 " 11

1. ² Γ ν2 (ν − 1) π ν 1 = ²

1 E(X |X ≤ q² ) = ²

Z

2

Z

q²

q²

(5)

x2 fX (x)dx

−∞

¡ ν+1 ¢ µ ¶ ν+1 2 − 2 Γ 1 x x2 ¡ ν2 ¢ √ dx 1+ ν νπ Γ 2 −∞ ¡ ¢ µ ¶− ν+1 Z q² +1 2 1 Γ ν+1 1 x2 ν 2 ¡ν ¢ √ = xd 1 + ² Γ 2 ν νπ 1 − ν −∞ Ã r ! ν ν−2 = q² E(X|X ≤ q² ) + Fν−2 q² , if ν > 2. ²(ν − 2) ν (6) 1 = ²

Z

q²

where the last equality follows by integration by parts and Fν (x) is the c.d.f. of Student’s t distribution with ν degrees of freedom. Plugging these expressions in (4), we obtain the expression for the variance σ²2 . Note that, besides an equation for σ²2 , we can explicitly calculate the AVaR of X since in the case of Student’s t distribution we can express AVaR as a conditional expectation, AV aR² (X) = −E(X|X ≤ q² ). Having an expression for the variance allows us to use the test of Kolmogorov and address the question oh how many simulations are needed in order to accept the hypothesis that the distribution of the random variable in the left-hand side of the limit relation (1), 6

472

STOYANOV-RACHEV

ν 3 4 5 6 7 8 9 10 15 25 50 ∞

² = 0.01 70000 60000 50000 23000 14000 13000 12000 12000 11000 10000 10000 10000

² = 0.05 17000 9000 7000 4500 4200 4100 4000 3900 3850 3800 3750 3300

Table I: The number of observations sufficient to accept the normal distribution as an approximate model for different values of ν and ². √ ³ ´ n \ AV aR² (X) − AV aR² (X) , (7) σ² is standard normal. If we accept the null hypothesis for a given value of n, then the standard normal distribution can be used as an approximate model and we can calculate not only confidence intervals but also other characteristics based on it. Table I shows the values of n sufficient to accept the null hypothesis in the test of Kolmogorov for different degrees of freedom and tail probabilities. We chose ² = 0.01 and ² = 0.05 since these values are frequently used in financial industry in value-at-risk estimation. The numbers in the table are calculated by generating independently 2000 samples of a given size and then from each sample (7) is estimated. In effect, we obtain 2000 observations from the distribution of (7). In line with intuition, the numbers Table I indicate that when the tail is heavier, we need a larger sample in order for the asymptotic law to be sufficiently close to the distribution of (7) in terms of the Kolmogorov metric. Another expected conclusion is that as the tail probability increases, a smaller sample turns out to be sufficient. In Table II, we calculated the 95% confidence interval for AVaR when the sample size changes from 250 to 10000 observations. We generated 2000 independent samples and then computed the quantity in equation (7). Thus, the 95% confidence intervals are obtained from 2000 observations of the random variable in (7). As n increases, the two quantiles approach the corresponding quantiles of the standard normal distribution. Note that the largest 7

ASYMPTOTIC DISTRIBUTION...

ν 3 4 5 6 7 8 9 10 15 25 50 ∞

n = 250 q2.5% q97.5% -1.110 2.011 -1.337 2.144 -1.441 2.153 -1.522 2.134 -1.627 2.050 -1.655 2.028 -1.720 1.938 -1.747 1.925 -1.813 1.751 -1.848 1.760 -1.898 1.948 -1.921 1.761

n = 500 q2.5% q97.5% -1.257 2.173 -1.442 2.229 -1.529 2.224 -1.618 2.033 -1.668 1.975 -1.760 2.145 -1.753 2.146 -1.809 1.980 -1.848 1.896 -1.933 2.028 -1.962 1.900 -1.976 1.920

n = 1000 q2.5% q97.5% -1.352 2.202 -1.543 2.082 -1.728 2.190 -1.701 2.115 -1.827 2.043 -1.836 2.032 -1.798 2.075 -1.762 2.078 -1.891 1.956 -1.897 1.950 -1.971 1.973 -1.964 1.822

473

n = 5000 q2.5% q97.5% -1.633 2.037 -1.744 2.230 -1.843 2.060 -1.848 1.987 -1.841 2.048 -1.898 2.034 -1.866 2.005 -1.822 1.950 -1.969 1.941 -1.939 1.957 -1.961 1.914 -1.869 1.907

n = 10000 q2.5% q97.5% -1.664 2.007 -1.756 2.176 -1.807 2.009 -1.955 1.982 -1.913 2.014 -1.866 1.939 -1.905 2.007 -1.962 2.000 -1.968 1.873 -1.899 1.923 -1.895 1.948 -2.004 1.937

Table II: The 95% confidence bounds generated from 2000 simulations from the distribution of (7) with ² = 0.01. The corresponding quantiles of N (0, 1) are q2.5% = −1.96 and q2.5% = 1.96.

n = 10000 is generally below the sample sizes for ² = 0.01 given in Table I. Nevertheless, the relative discrepancies between the quantiles given in Table II and the corresponding standard normal distribution quantiles are less than 5% for ν ≥ 6.1 The relative discrepancies between the quantiles given in Table III the corresponding standard normal distribution quantiles for n = 10000 have the same magnitude. However, in this case n = 10000 is well above the sample sizes given in Table I for ² = 0.05. As a result, we can conclude that even smaller samples than the ones given in Table I can lead to 95% confidence intervals obtained via resampling from (7) being close to the corresponding 95% confidence interval obtained from the limit distribution even though the Kolmogorov test fails for such samples. For instance, the relative deviation between the quantiles given in Table II for n = 5000 and the corresponding standard normal distribution quantiles are below 7% for n ≥ 6, which is a small deviation for all practical purposes. As a result of this analysis, we can conclude that for the purposes of building confidence intervals for AV aR² (X) when X ∈ t(ν), with ν ≥ 6 and ² = 0.01, 0.05, we can safely employ the asymptotic law when the sample size we use for AVaR estimation contains more than 5000 observations. If Student’s t distribution is fitted on daily stock-returns time series, such values 1

If we generate a sample of 2000 observations from the standard normal distribution, a relative deviation below 6% between the estimated quantile q2.5% and the corresponding standard normal quantile happens with about 95% probability, and below 7.7% with about 99% probability.

8

474

STOYANOV-RACHEV

ν 3 4 5 6 7 8 9 10 15 25 50 ∞

n = 250 q2.5% q97.5% -1.422 2.110 -1.647 2.169 -1.749 2.081 -1.810 2.071 -1.786 2.215 -1.932 2.131 -1.848 2.139 -1.906 2.103 -1.797 1.905 -1.958 1.950 -1.986 1.927 -2.013 1.828

n = 500 q2.5% q97.5% -1.543 2.016 -1.737 2.235 -1.811 2.096 -1.896 2.030 -1.824 1.990 -1.870 2.058 -1.884 2.081 -2.021 1.966 -1.929 2.056 -1.994 1.956 -1.980 1.823 -1.953 1.869

n = 1000 q2.5% q97.5% -1.549 1.981 -1.787 2.171 -1.757 2.148 -1.921 1.941 -1.809 2.086 -1.755 2.090 -1.930 2.023 -1.839 2.087 -1.944 1.952 -1.939 1.968 -1.962 1.883 -1.975 1.893

n = 5000 q2.5% q97.5% -1.725 1.947 -1.900 2.226 -1.868 2.015 -1.958 1.998 -1.986 2.030 -1.937 2.014 -1.995 1.964 -2.009 1.930 -1.924 1.973 -2.085 1.993 -1.911 1.969 -2.034 1.958

n = 10000 q2.5% q97.5% -1.883 1.987 -1.849 2.115 -1.937 2.100 -1.886 2.032 -1.916 2.015 -1.915 1.952 -1.863 2.048 -1.989 1.995 -1.947 1.979 -1.894 1.944 -2.002 1.935 -1.903 1.944

Table III: The 95% confidence bounds generated from 2000 simulations from the distribution of (7) with ² = 0.05. The corresponding quantiles of N (0, 1) are q2.5% = −1.96 and q2.5% = 1.96.

for ν are very common. Figure 1 illustrates the differences in the convergence rate when X has Student’s t distribution with ν = 3, which corresponds to heavier tails, and ν = 10. Since high degrees of freedom imply more light tails, smaller samples are sufficient for the density of (7) to be closer to the standard normal density.

3.2

The effect of tail truncation

The stochastic stability of sample AVaR increases dramatically after tail truncation. In this section, we repeat the calculations from the previous section but when X has Student’s t distribution with the tails truncated at q0.1% and q99.9% quantiles. The random variable Y is said to have a truncated distribution at these quantiles if it has the representation Y = XI{q0.1% ≤ X ≤ q99.9% } + q0.1% I{X < q0.1% } + q99.9% I{X > q99.9% } in which X ∈ t(ν), I{A} denotes the indicator of the event A, and q0.1% , q99.9% are the corresponding quantiles of X. The tail truncation introduces small point masses at the two quantile levels. The two conditional expectations in (4) can be related to the corresponding conditional expectations of X. In the following, we assume that the tail probability ² is larger from the tail probability of the left truncation point, 9

ASYMPTOTIC DISTRIBUTION...

475

0.8 n = 250 n = 1000 n = 10000 n = 70000 N(0,1)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −5

0

5

0.8 n = 250 n = 1000 n = 10000 n = 12000 N(0,1)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −5

0

5

Figure 1: The density of (7) approaching the N (0, 1) density as the sample size increases with ν = 3 (top) and ν = 10 (bottom).

10

476

STOYANOV-RACHEV

ν 3 4 5 6 7 8 9 10 15 25 50 ∞

² = 0.01 12000 11500 11000 11000 10500 10000 10000 10000 10000 10000 10000 10000

² = 0.05 4000 3600 3300 3200 3100 3000 3000 3000 2950 2900 2900 2900

Table IV: The number of observations sufficient to accept the normal distribution as an approximate model for different values of ν and ². ² > 0.001. Under this assumption, the ²-quantile of X is the same as the ²-quantile of Y .

E(Y |Y ≤ q² ) = E(X|X ≤ q² ) −

0.001 0.001q² E(X|X ≤ q0.1% ) + ² ²

0.001 0.001q²2 E(X 2 |X ≤ q0.1% ) + ² ² in which the conditional expectations of X can be computed according to formulae (5) and (6). Plugging the expressions for the conditional expectations of Y in the expression for σ²2 , we obtain the variance of the asymptotic distribution. Furthermore, the tail truncation does not break the link between AVaR and the conditional expectation, therefore E(Y 2 |Y ≤ q² ) = E(X 2 |X ≤ q² ) −

AV aR² (Y ) = −E(Y |Y ≤ q² ). In the following, we investigate the convergence rate of √ ³ ´ n \ AV aR² (Y ) − AV aR² (Y ) , (8) σ² for different degrees of freedom to the standard normal distribution and we compare the results to the ones in the previous section. Table IV is the counterpart of Table I for the truncated distribution. It is impressive how the sample size sufficient to accept the null hypothesis in the Kolmogorov test decreases after tail truncation. The most dramatic change 11

ASYMPTOTIC DISTRIBUTION...

ν 3 4 5 6 7 8 9 10 15 25 50

n = 250 q2.5% q97.5% -1.723 1.699 -1.759 1.694 -1.808 1.536 -1.947 1.565 -1.960 1.524 -2.002 1.567 -1.963 1.552 -2.003 1.596 -2.090 1.485 -2.183 1.502 -2.272 1.509

n = 500 q2.5% q97.5% -1.847 1.932 -1.863 1.819 -1.884 1.871 -1.937 1.759 -1.960 1.666 -2.015 1.693 -2.030 1.748 -2.119 1.709 -2.159 1.650 -2.084 1.578 -2.089 1.632

n = 1000 q2.5% q97.5% -1.850 1.958 -1.903 1.860 -1.926 1.932 -2.002 1.734 -1.965 1.844 -1.903 1.802 -2.106 1.779 -2.034 1.850 -2.065 1.786 -2.093 1.747 -2.042 1.726

477

n = 5000 q2.5% q97.5% -1.966 1.921 -1.989 1.942 -1.961 1.964 -2.057 1.946 -2.101 1.932 -1.952 1.856 -1.965 2.026 -1.925 1.813 -1.983 1.847 -2.016 1.806 -1.938 1.914

n = 10000 q2.5% q97.5% -1.860 1.936 -1.964 1.886 -1.782 2.066 -1.981 1.958 -1.981 1.927 -1.917 1.928 -1.932 1.938 -1.990 1.952 -2.035 1.855 -1.954 1.877 -2.056 1.970

Table V: The 95% confidence bounds generated from 2000 simulations from the distribution of (8) with ² = 0.01. The corresponding quantiles of N (0, 1) are q2.5% = −1.96 and q2.5% = 1.96.

is in the case ν = 3. Now we need only 12000 observations compared to 70000 in the non-truncated case. Tables V and VI are the counterparts of Tables II and III. The relative deviation of the quantiles q2.5% and q97.5% of the random variable in (8) from those of the standard normal distribution are below 7% for all degrees of freedom and n = 10000, and, with a few exceptions, for n = 5000. Compare Figure 2 and the top plot in Figure 1 for an illustration of the improvement in the convergence rate. These results indicate that the asymptotic distribution can be used to obtain a 95% confidence bound for the sample AVaR for all degrees of freedom if the sample size contains more than 5000 observations.

3.3

Infinite variance distributions

A critical assumption behind the limit result in Theorem 1 is the finite variance of X. To be more precise, the condition of finite variance can be loosened to finite downside semi-variance, D max(−X, 0) < ∞, because it is the behavior of the left tail which is important. As a consequence, the sample AVaR of distributions with infinite variance, but finite downside semi-variance, may still follow Theorem 1. However, there are infinite variance distributions for which D max(−X, 0) = ∞ 12

478

STOYANOV-RACHEV

ν 3 4 5 6 7 8 9 10 15 25 50

n = 250 q2.5% q97.5% -1.815 2.116 -1.756 2.150 -1.820 1.954 -1.899 2.089 -2.001 2.032 -1.888 1.995 -2.017 2.003 -1.928 1.814 -2.059 1.983 -1.999 1.854 -1.960 1.898

n = 500 q2.5% q97.5% -1.866 2.041 -1.811 2.073 -1.971 2.032 -1.981 2.036 -1.921 1.997 -1.922 2.050 -1.892 1.918 -1.992 1.960 -2.020 2.007 -2.038 1.945 -2.028 1.898

n = 1000 q2.5% q97.5% -1.939 2.018 -2.052 2.060 -1.916 2.036 -2.012 2.012 -1.949 1.980 -1.907 1.917 -1.899 2.017 -1.870 1.949 -1.961 1.922 -1.889 2.028 -1.947 1.906

n = 5000 q2.5% q97.5% -1.944 1.975 -1.923 1.973 -1.826 1.960 -1.955 1.933 -1.980 1.936 -1.942 1.911 -1.931 2.001 -1.845 2.076 -1.953 1.870 -2.031 1.916 -2.015 2.002

n = 10000 q2.5% q97.5% -2.045 1.874 -1.922 1.854 -1.941 1.883 -1.921 2.011 -2.016 1.915 -1.910 1.903 -2.009 1.967 -1.992 1.898 -1.936 1.874 -1.975 1.890 -1.959 1.911

Table VI: The 95% confidence bounds generated from 2000 simulations from the distribution of (8) with ² = 0.05. The corresponding quantiles of N (0, 1) are q2.5% = −1.96 and q2.5% = 1.96.

0.8 n = 250 n = 1000 n = 5000 n = 10000 N(0,1)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −5

0

5

Figure 2: The density of (8) approaching the N (0, 1) density as the sample size increases with ν = 3 and ² = 0.01.

13

ASYMPTOTIC DISTRIBUTION...

479

0.8 n = 250 n = 1000 n = 5000 n = 10000 N(0,1)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −5

0

5

Figure 3: Lack of convergence, X has a stable distribution with X ∈ S1.5 (1, 0, 0) and ² = 0.05.

and, therefore, the limit result in Theorem 1 does not hold for them. Such is the class of stable distributions which arises from generalizations of the Central Limit Theorem and has been proposed as a model for stock return distributions, see Rachev and Mittnik (2000). Stable distributions are introduced by their characteristic functions. X is said to have a stable distribution if its characteristic function is ( ϕ(t) = EeitX =

exp{−σ α |t|α (1 − iβ |t|t tan( πα )) + iµt}, α 6= 1 2 2 t exp{−σ|t|(1 + iβ π |t| ln(|t|)) + iµt}, α=1

Except for a couple of representatives, generally no closed-form expressions for their densities and c.d.f.s are known. If α < 2, then X has infinite variance. If 1 < α ≤ 2, then X has finite mean and the AVaR of X can be calculated. In our calculations, we will use the semi-analytic formula in Stoyanov et al. (2006). Even though we know that Theorem 1 does not hold for a stable distribution with α < 2, we simulate 2000 draws from the random variable 14

480

STOYANOV-RACHEV

0.8 n = 250 n = 1000 n = 5000 n = 10000 N(0,1)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −5

0

5

Figure 4: After tail truncation at q0.1% and q99.9% , there is a fast convergence to N (0, 1), α = 1.5 and ² = 0.05.

in equation (7) in which σ² is estimated from a generated sample by estimating the corresponding conditional moments. In theory these the second conditional moment explodes but for any finite sample its estimate is a finite number. Our goal is to see what happens when Theorem 1 does not hold. Figure 3 illustrates such a divergent case in which α = 1.5 and ² = 0.05. The lack of convergence is quite obvious. Stable distributions with α < 2 in combination with a tail truncation method have been proposed as a model for the returns of the underlying in derivatives pricing. It is interesting to see how much the simple truncation technique we applied in the previous section can change Figure 3. With its tails truncated according to our simple method, the random variable becomes with a bounded support and, therefore, it has finite variance. As a consequence, Theorem 1 holds. Figure 4 illustrates this change. We observe a quick convergence rate, similar to the one illustrated in Figure 2 for Student’s t distribution.

15

ASYMPTOTIC DISTRIBUTION...

4

Conclusion

In this paper, we study the asymptotic distribution of sample AVaR. Under certain regularity conditions, we prove a limit theorem in which the limiting distribution is the normal distribution. We study how the convergence rate in the limit theorem is influenced by the tail behavior of the random variable. An expected result is that, other things equal, more observations are needed when the tail is heavier. We find out that a simple tail truncation method improves dramatically the convergence rate. As a consequence, the asymptotic distribution is reliable for confidence interval calculations when the number of simulations is more than 5000 if the random variable has a truncated Student’s t distribution. We also consider an infinite variance case in which the random variable as a stable distribution with finite mean. We illustrate the lack of convergence and demonstrate the improvement due to tail truncation at high quantiles.

16

481

482

STOYANOV-RACHEV

References Artzner, P., F. Delbaen, J.-M. Eber and D. Heath (1998), ‘Coherent measures of risk’, Math. Fin. 6, 203–228. Biglova, A., S. Ortobelli, S. Rachev and S. Stoyanov (2004), ‘Different approaches to risk estimation in portfolio theory’, The Journal of Portfolio Management Fall 2004, 103–112. Pflug, G. (2000), ‘Some remarks on the value-at-risk and the conditional value-at-risk’, In: Uryasev, S. (Ed.), Probabilistic Constrained Optimization: Methodology and Applications. Kluwer Academic Publishers, Dordrecht . Rachev, S., D. Martin, B. Racheva-Iotova and S. Stoyanov (2006), ‘Stable etl optimal portfolios and extreme risk management’, forthcoming in Decisions in Banking and Finance, Springer/Physika, 2007 . Rachev, S., F. Fabozzi and C. Menn (2005), Fat-tails and skewed asset return distributions, Wiley, Finance. Rachev, S. T., Stoyan V. Stoyanov and F. J. Fabozzi (2008), Advanced stochastic models, risk assessment, and portfolio optimization: The ideal risk, uncertainty, and performance measures, Wiley, Finance. Rachev, S.T. and S. Mittnik (2000), Stable Paretian Models in Finance, John Wiley & Sons, Series in Financial Economics. Stoyanov, S., G. Samorodnitsky, S. Rachev and S. Ortobelli (2006), ‘Computing the portfolio conditional value-at-risk in the α-stable case’, Probability and Mathematical Statistics 26, 1–22. Stoyanov, S., S. Rachev and F. Fabozzi (2007), ‘Optimal financial portfolios’, forthcoming in Applied Mathematical Finance .

17

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.4,483-493,2008, COPYRIGHT 2008 EUDOXUS PRESS, 483 LLC

A note on generalized twisted q-Euler numbers and polynomials C. S. Ryoo†∗, J.J. Seo† , T. Kim‡ †

Department of Mathematics, Hannam University, Daejeon 306-791, Korea e-mail: [email protected] ‡

EECS, Kyungpook National University, Taegu, 702-701, Korea e-mail: [email protected] Abstract In this paper we construct a new generalized twisted q-Euler polynomials and generalized twisted q-Euler numbers attached to χ. We investigate some properties which are related to the generalized twisted q-Euler Polynomials. We also derive the existence of a specific interpolation function which interpolate the generalized twisted q- Euler polynomials at negative integer. 2000 Mathematics Subject Classification - 11B68, 11S40 Key words- Euler numbers, Euler polynomials, Generalized Euler numbers, Generalized Euler polynomials, Euler numbers, Euler polynomials, Generalized twisted q-Euler numbers, Generalized twisted q-Euler polynomials

1.

Introduction

Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assume 1 that |q| < 1. If q ∈ Cp , we normally assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. [x]q = [x : q] = ∗ This

1 − qx , cf. [1, 2, 3, 4, 5, 9, 11] . 1−q

work was supported by Hannam University Research Fund, 2007

1

484

RYOO ET AL

Hence, limq→1 [x] = x for any x with |x|p ≤ 1 in the present p-adic case. For g ∈ U D(Zp , Cp ) = {g|g : Zp → Cp is uniformly differentiable function}, the p-adic q-integral (or q-Volkenborn integration ) was defined by [3] Z Iq (g) =

g(x)dµq (x) = lim

1

N →∞

Zp

[pN ]q

N pX −1

q x g(x).

x=0

Let d be a fixed integer and let p be a fixed prime number. For any positive integer N , we set X = Xd = lim(Z/dpN Z), ←− N [ ∗ X = (a + dpZp ),

X1 = Zp ,

0 1, the Frobenius-Euler numbers Hn (u) belonging to u are defined by the generating function eH(u)t =

∞ X 1−u tn = Hn (u) , cf. [1,4,9,10], t e − u n=0 n!

with the usual convention of symbolically replacing H n by Hn . The Euler polynomials En (x) are defined by eE(x)t =

∞ X 2 tn xt e = E (x) , cf. [3,4,5,8,9,10]. n et + 1 n! n=0

2

GENERALIZED TWISTED Q-EULER NUMBERS...

485

For u ∈ C with |u| > 1, the Frobenius-Euler polynomials Hn (u, x) belonging to u are defined by eH(u,x)t =

∞ 1 − u xt X tn e = H (u, x) , cf. [8, 9, 10]. n et − u n! n=0

T. Kim[2] gave relation between Bn,w and Hn (u), nth Euler numbers as follows: Bn,w =

n Hn−1 (w−1 ) if w 6= 1. w−1

In [2], Kim defined the locally constant function as follows: Let Tp = ∪m≥1 Cpm = lim Cpm , m→∞

m

where Cpm = {w|wp = 1} is the cyclic group of order pm . For w ∈ Tp , we denote by φw : Zp → Cp the locally constant function x 7−→ wx . If we take g(x) = φw (x)etx , then we easily see that Z t φw (x)etx dµ1 (x) = . t−1 we Zp T.Kim [2] treated analogue of Bernoulli numbers, which is called twisted Bernoulli numbers. We define the twisted Bernoulli polynomials Bn,w (x) ext

∞ X t tn = Bn,w (x) . t we − 1 n=0 n!

By using Taylor series of etx in the above equation, we obtain Z xn φw (x)dµ1 (x) = Bn,w , Zp

where Bn,w = Bn,w (0). Now, we consider the case q ∈ (−1, 0) corresponding to q-deformed fermionic certain and annihilation operators and the literature given therein [3,4,5,7,8]. The expression for the Iq (g) remains same, so it is tempting to consider the limit q → −1. That is, Z X I−1 (g) = lim Iq (g) = g(x)dµ−1 (x) = lim g(x)(−1)x . (1.1) q→−1

N →∞

Zp

0≤x 0 . Since (2.3) is uniformly asymptotically stable and Bi are continuous in t , it follows that GmΤ (t1 , t0 )W −1 (t0 , t1 ) ≤ k1 , for some k1 > 0 , T (t1 , t0 )φ (0) ≤ k2 exp ( −α (t1 − t0 ) ) , for some k2 > 0 , H (t ,η ) ≤ k3 exp ( −α (t1 − t0 ) ) , for some k3 > 0 .

Hence, t1 u (t ) ≤ k1 ⎡ k2 exp ( −α (t1 − t0 ) ) + k3 + ∫ M exp ( −α (t1 − s) ) exp(− β s )π ( x (⋅), u (⋅) ) ds ⎤ , t0 ⎢⎣ ⎥⎦

8

NULL CONTROLLABILITY...

517

and therefore u (t ) ≤ k1 ⎡⎣(k2 + k3 ) exp ( −α (t1 − t0 ) ) + KM exp( −α t1 ) ⎤⎦

(3.4)

since β − α ≥ 0 and s ≥ t0 ≥ 0 . Hence, by taking t1 sufficiently large, we have u (t ) ≤ a , t ∈ [t0 , t1 ] which proves that u is an admissible control. We now prove the existence of a solution pair of the integral equations (3.1) and (3.2). Let B be the Banach space of all functions ( x, u ) :[t0 − h, t1 ] × [t0 − h, t1 ] → R n × R m

where x ∈ B ([t0 − h, t1 ], R n ) and u ∈ L2 ([t0 − h, t1 ], R m ) with the norm defined by ( x, u ) ≤ x 2 + u 2 ,

where x 2 =

{∫

t1

t0 − h

2

}

1 2

x( s ) ds , and u 2 =

{∫

t1

t0 − h

2

}

1 2

u ( s ) ds .

Define the operator T : B → B by T ( x, u ) = ( y , v) , where v(t ) = −GmΤ (t1 , t )W −1 (t0 , t1 ) q (t1 ,η ) for t ∈ [t0 , t1 ]

(3.5)

and v(t ) = η (t ) for t ∈ [t0 − γ , t0 ] ; t

t

t0

t0

y (t ) = T (t , t0 )φ (0) + H (t ,η ) + ∫ Gm (t , s )v( s )ds + ∫ X (t , s ) t

(∫

0

−γ

+ ∫ X (t , s ) f ( s, xs , u ( s ) )ds for t ∈ J t0

)

A(θ ) x (t + θ )dθ ds

(3.6)

and y (t ) = φ (t ) for t ∈ [t0 − γ , t0 ] . From equation (3.4), we have shown that v(t ) ≤ a , t ∈ J and also v :[t0 − h, t0 ] → IU , so v(t ) ≤ a . Hence,

9

518

UMANA

1

v(t ) 2 ≤ a (t1 + h − t0 ) 2 = β 0 . Also t

y (t ) ≤ k2 + k3 exp ( −α (t − t0 ) ) + k 4 ∫ v (s) ds + KM exp( −α t1 ) t0

where k4 = sup Gm (t , s ) . Since α > 0 , t ≥ t0 ≥ 0 , it follows that y (t ) ≤ k2 + k3 + k4 a (t1 − t0 ) + KM ≡ β , t ∈ J

and y (t ) ≤ sup φ (t ) ≡ δ , t ∈ [t0 − h, t0 ] .

Hence, if λ = max {β , δ } , then 1

y 2 ≤ λ (t1 + h − t0 ) 2 ≡ β1 < ∞ . Let r = max {β 0 , β1} . Then letting Q (r ) = {( x, u ) ∈ B : x 2 ≤ r , u 2 ≤ r} , it follows that T : Q (r ) → Q(r ) . Since Q(r ) is closed, bounded and convex, by Riesz’s theorem, it is relatively compact under the transformation T . The Schauder fixed point theorem implies that T has a fixed point ( x, u ) ∈ Q(r ) . This fixed point ( x, u ) of T is a solution pair of the integral equations (3.5), (3.6). Hence, the system (1.8) is Euclidean null controllable. 4. Applications If we now specialize to the constant systems with multiple delays in the control defined by d ( x (t ) − A−1 x(t − h) ) = A0 x(t ) + A1 x (t − h) + B0u (t ) + B1u (t − h) dt

(4.1)

0 d ( x (t ) − A−1 x(t − h) ) = A0 x(t ) + A1 x (t − h) + B0u (t ) + B1 (t − h) +C0 ∫−∞ exp(ηθ ) x (t + θ )dθ (4.2) dt

10

NULL CONTROLLABILITY...

519

d ( x (t ) − A−1 x(t − h) ) = A0 x(t ) + A1 x (t − h) + B0u (t ) + B1u (t − h) dt +C0 ∫ exp(ηθ ) x (t + θ ) dθ + f ( t , x(t ), x (t − h ), u (t ) ) 0

−∞

(4.3)

then the following results follow: Theorem 4.1: If rank [ B0 , A0 B0 ] = n , then system (4.1) is completely controllable on [t0 , t1 ] . Proof: This is equivalent to Theorem 2 of Gahl [9]. Theorem 4.2: In system (4.2), assume that (i) (4.2) with u = 0 is uniformly asymptotically stable, (ii) rank [ B0 , A0 B0 ] = n , then system (4.2) is null controllable with constraints. Proof: By (ii), (4.1) is completely controllable. Hence (i) and (ii) satisfy the requirements of Theorem 3.1 and the proof is complete. Theorem 4.3: For system (4.3), assume that (i) f satisfies all smoothness conditions for the existence and uniqueness of solutions, (ii) the zero solution of (4.2) with u = 0 is uniformly asymptotically stable, (iii) rank [ B0 , A0 B0 ] = n , (iv) f (t ,0, 0, 0) = 0 , then system (4.3) is null controllable with constraints. Proof: Immediate from Theorems 3.1 and 4.1. Example Consider the system d ( x (t ) − A−1 x(t − h) ) = A0 x(t ) + A1 x (t − h) + B0u (t ) + B1u (t − h) dt

11

520

UMANA

+C0 ∫ exp(ηθ ) x (t + θ ) dθ + f ( t , x(t ), x (t − h ), u (t ) ) 0

−∞

(4.4)

where ⎛0 1⎞ ⎛ −1 1 ⎞ ⎛0 3 ⎞ ⎛ 0⎞ ⎛ −1⎞ ⎛0 0 ⎞ A−1 = ⎜ ⎟ , A0 = ⎜ ⎟ , A1 = ⎜ ⎟ , B0 = ⎜ ⎟ , B1 = ⎜ ⎟ , C0 = ⎜ ⎟ ⎝1 0⎠ ⎝ 1 −2 ⎠ ⎝ 0 −1 ⎠ ⎝1⎠ ⎝0⎠ ⎝ 0 −1 ⎠ with 0 ⎛ ⎞ f ( t , x (t ), x(t − h ), u (t ) ) = ⎜ − t ⎟. ⎝ e sin ( x(t ) + x(t − h) ) cos u (t ) ⎠

The characteristic roots of the homogeneous equation 0 d ( x (t ) − A−1 x(t − h) ) = A0 x(t ) + A1 x(t − h) + ∫−∞ exp(ηθ ) x(t + θ )dθ dt

(4.5)

is 0

λ 2 + 3λ + 1 + (3λ − λ 2 )e −2 λ + (2 − 3λ ) e− λ + (λ + 1) ∫ exp[( λ + η )θ ]dθ = 0 −∞

(4.6)

Every root of (4.6) has negative real part. Hence, by Theorem 1 of Sinha [15], system (4.5) is uniformly asymptotically stable. We now show that the linear base system d ( x (t ) − A−1 x(t − h) ) = A0 x(t ) + A1 x (t − h) + B0u (t ) + B1u (t − h) dt

(4.7)

is controllable on any interval [0, t ] , t > 0 . By Theorem 4.1, we show that rank [ B0 , A0 B0 ] = n . But ⎛0 1 ⎞ ⎛0 1⎞ rank [ B0 , A0 B0 ] = rank ⎜ = rank ⎜ ⎟ ⎟=2=n. ⎝ 1 −2 ⎠ ⎝1 0⎠ Since rank [ B0 , A0 B0 ] = 2 for each t > 0 , the system (4.7) is controllable on each [0, t ] , t > 0 on R n . We conclude that system (4.4) is null controllable, by Theorem 3.1, since

12

NULL CONTROLLABILITY...

521

f ( t , x (t ), x(t − h), 0 ) ≤ e− t sin ( x(t ) + x(t − h) ) ≤ e − t ≡ π (t ) . Conclusion We have derived sufficient conditions for the null controllability of nonlinear infinite neutral systems with time varying multiple delays in control. These conditions are given with respect to the uniform asymptotic stability of the free linear base system and the controllability of the linear controllable base system, with the assumption that the perturbation function f satisfies some smoothness and growth conditions. References [1] R. Brayton, Nonlinear oscillations in a distributed network, Quart. Appl. Math. 24, 239-301 (1976). [2] K. Balachandran, J. P. Dauer and P. Balasubramanian, Local null controllability of nonlinear functional differential systems in Banach spaces, J. Optim. Theory Appl. 88, 61-75 (1995). [3] K. Balachandran and J. P. Dauer, Null controllability of nonlinear infinite delay systems with distributed delays in control, J. Math. Anal. Appl. 145, 274-281 (1990). [4] K. Balachandran and J. P. Dauer, Null controllability of nonlinear infinite delay systems with time varying multiple delays in control, Appl. Math. Letters 9, 115-121 (1996). [5] K. Balachandran and E. R. Anandhi, Controllability of neutral functional integrodifferential infinite delay systems in Banach spaces, Taiwanese J. Math. 8, 689-702 (2004). [6] K. Balachandran and A. Leelamani, Null controllability of neutral evolution integrodifferential systems with infinite delay, Math. Prob. Engineering 2006, 1-18 (2006). [7] E. N. Chukwu, On the null controllability of nonlinear delay systems with restrained controls, J. Math. Anal. Appl. 76, 283-296 (1980).

13

522

UMANA

[8] X. Fu, Controllability of neutral functional differential systems in abstract space, Applied Math. Comp. 141, 281-296 (2003). [9] R. D. Gahl, Controllability of nonlinear systems of neutral type, J. Math. Anal. Appl. 66, 3342 (1978). [10] H. Hermes and J. P. Laselle, Functional Analysis and Time Optimal Control, Academic Press, New York, 1969. [11] V. A. Iheagwam and J. U. Onwuatu, Relative controllability and null controllability of linear delay systems with distributed delays in the state and control, J. Nigerian Asso. Math. Physics 9, 221-238 (2005). [12] D. Iyai, Euclidean null controllability of infinite neutral differential systems, ANZIAM J. 48, 285-293 (2006). [13] J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrect, 1991. [14] J. U. Onwuatu, Null controllability of nonlinear infinite neutral system, Kybernetika 29, 325-336 (1993). [15] A. S. C. Sinha, Null controllability of nonlinear infinite delay systems with restrained controls, Int. J. Control 42, 735-741 (1985). [16] R. A. Umana, Relative null controllability of linear systems with multiple delays in state and control, J. Nigerian Asso. Math. Physics 10, 517-522 (2006). [17] R. A. Umana and C. A. Nse, Null controllability of nonlinear integrodifferential systems with delays, Journal of Advances in Modelling 61, 73-84 (2006).

14

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.4,523-525,2008, COPYRIGHT 2008 EUDOXUS PRESS, 523 LLC

Moments of the Scaled Burr Type X Distribution by M.

Zhou1 ,

D.

Yang2 ,

Y. Wang3 & S. Nadarajah4

Abstract: A recent paper by Surles and Padgett proposed the scaled Burr type X distribution and discussed various properties, including moments. The paper claimed that closed-form expressions for E(X k ) are possible only for certain special cases: when the parameter of the distribution is assumed to be an integer or when k = 2 (the latter represented as an infinite sum). In this note, we show that one can derive simple expressions for E(X k ) for all even k ≥ 2 without any restriction on the parameter of the distribution. The expressions only involve the gamma function and its derivatives.

1

Introduction

Surles and Padgett (2005) defined the scaled Burr X distribution with shape parameter θ and scale parameter σ by the cdf     θ x 2 F (x) = 1 − exp − (1) σ for x > 0, θ > 0 and σ > 0. Surles and Padgett (2005) discussed various properties of this distribution, including moments and their approximations, maximum likelihood estimators and their asymptotic properties as well as types I and II censoring. This distribution is a particular case of the exponentiated Weibull distribution introduced by Mudholkar et al. (1995); see Mudholkar and Hutson (1996), Nassar and Eissa (2003) and Nadarajah and Gupta (2005) for more recent developments. This note concerns the moment properties of a random variable X having the cdf (1). Surles and Padgett (2005) claim that ‘closed-form expressions for the moments only exist for certain special cases . . ..” In particular, two closed-form expressions are given: 

E X

k



k



= σ θΓ

X   θ−1 k 1 j θ−1 +1 (−1) 2 j (j + 1)k/2+1 j=0

applicable when θ ≥ 1 is an integer; and, E X2




= θσ 2

∞ X i=0

1

1 i(θ + i)

Author’s address: Department of Statistics, University of Nebraska, Lincoln, Nebraska 68583, USA Author’s address: Department of Statistics, University of Nebraska, Lincoln, Nebraska 68583, USA 3 Author’s address: Department of Statistics, University of Nebraska, Lincoln, Nebraska 68583, USA 4 Corresponding author’s address: School of Mathematics, University of Manchester, Manchester M60 1QD, UK, E-mail: [email protected] 2

1

524

ZHOU ET AL

applicable for any θ > 0. In this note, we show that one can derive simple expressions for E(X k ) for any even k ≥ 2 and for any θ > 0. The expressions only involve the gamma function and its derivatives.

2

Moments

Theorem 1 derives the expression for the kth moment for any even k ≥ 2 and for any θ > 0. Theorem 1 If X is a random variable with the cdf (1) then 

E X

k



k

=

k/2

θΓ(θ)σ (−1)

∂ k/2 ∂β k/2



Γ(β + 1) Γ(θ + β + 1)



(2) β=0

for any even k ≥ 2 and for any θ > 0. Proof: The pdf corresponding to (1) is: f (x) =

       θ−1 2θx x 2 x 2 1 − exp − 2 exp − σ σ σ

and so one can express E(X k ) as 

E X

k



=

2θ σ2

       θ−1 x 2 x 2 exp − 1 − exp − dx σ σ

∞

Z

k+1

x 0

k/2 k

= (−1)

Z

σ θ

1

y θ−1 {log(1 − y)}k/2 dy,

(3)

0

which follows by substituting y = 1 − exp{−(x/θ)2 }. The result in (2) follows by applying equation (2.6.9.5) in Prudnikov et al (1986, volume 1) to calculate the integral in (3).  The result in (2) can be used to derive simple expressions for moments of even-order. Corollary 1 illustrates this for the first five even-order moments. Corollary 1 If X is a random variable with the cdf (1) then the first five even-order moments are given by " #
E X 2 = σ 2 γ + Ψ (θ + 1) , " E X


4

E X


6

# 0

= (1/6)σ 4 π 2 − 6Ψ (θ + 1) + 6γ 2 + 12γΨ (θ + 1) + 6Ψ2 (θ + 1) , " 00

0

= (1/2)σ 6 4ζ (3) + 2Ψ (θ + 1) + π 2 γ + π 2 Ψ (θ + 1) − 6Ψ (θ + 1) γ # 0

3

2

2

3

−6Ψ (θ + 1) Ψ (θ + 1) + 2γ + 6γ Ψ (θ + 1) + 6γΨ (θ + 1) + 2Ψ (θ + 1) , " E X

8




= (1/20)σ

8

00

3π 4 − 20Ψ (3θ + 1) + 160ζ (3) γ + 160ζ (3) Ψ (θ + 1) + 80Ψ (θ + 1) γ

2

X DISTRIBUTION

525

n 0 o2 00 0 +80Ψ (θ + 1) Ψ (θ + 1) − 20π 2 Ψ (θ + 1) + 60 Ψ (θ + 1) + 20π 2 γ 2 0

+40π 2 γΨ (θ + 1) + 20π 2 Ψ2 (θ + 1) − 120Ψ (θ + 1) γ 2 0

0

−240Ψ (θ + 1) γΨ (θ + 1) − 120Ψ (θ + 1) Ψ2 (θ + 1) + 20γ 4 + 80γ 3 Ψ (θ + 1) # +120γ 2 Ψ2 (θ + 1) + 80γΨ3 (θ + 1) + 20Ψ4 (θ + 1) , " E X

10




= (1/12)σ

00

10

0

60π 2 γ 2 Ψ (θ + 1) − 120Ψ (θ + 1) Ψ (θ + 1) + 240ζ (3) γ 2

n 0 o2 0 −60π 2 Ψ (θ + 1) Ψ (θ + 1) + 60π 2 γΨ2 (θ + 1) + 180 Ψ (θ + 1) Ψ (θ + 1) n 0 o2 0 00 +20π 2 γ 3 − 60π 2 Ψ (θ + 1) γ + 120Ψ (θ + 1) Ψ2 (θ + 1) + 180 Ψ (θ + 1) γ +480ζ (3) γΨ (θ + 1) + 60γ 4 Ψ (θ + 1) + 120γ 3 Ψ2 (θ + 1) + 120γ 2 Ψ3 (θ + 1) 0

+60γΨ4 (θ + 1) + 9π 4 γ + 9π 4 Ψ (θ + 1) + 40ζ (3) π 2 − 240ζ (3) Ψ (θ + 1) 0

00

−360Ψ (θ + 1) γ 2 Ψ (θ + 1) + 240ζ (3) Ψ2 (θ + 1) + 120Ψ (θ + 1) γ 2 00

0

+240Ψ (θ + 1) γΨ (θ + 1) + 20π 2 Ψ3 (θ + 1) − 120Ψ (θ + 1) γ 3 0

000

−120Ψ (θ + 1) Ψ3 (θ + 1) + 12Ψ5 (θ + 1) + 12γ 5 − 60Ψ (θ + 1) γ 000

00

0

−60Ψ (θ + 1) Ψ (θ + 1) + 20Ψ (θ + 1) π 2 − 360Ψ (θ + 1) γΨ2 (θ + 1) # 0000

+12Ψ

(θ + 1) + 288ζ (5) ,

where γP = 0.5772156649 · · · is Euler’s constant, Ψ(x) = d log Γ(x)/dx is the digamma function and −x is the zeta function. ζ(x) = ∞ k=1 k

References Mudholkar, G. S. and Hutson, A. D. (1996). The exponentiated Weibull family: some properties and a flood data application. Communications in Statistics—Theory and Methods, 25, 30593083. Mudholkar, G. S., Srivastava, D. K. and Freimer, M. (1995). The exponentiated Weibull family. Technometrics, 37, 436–445. Nadarajah, S. and Gupta, A. K. (2005). On the moments of the exponentiated Weibull distribution. Communications in Statistics—Theory and Methods, 34. Nassar, M. M. and Eissa, F. H. (2003). On the exponentiated Weibull distribution. Communications in Statistics—Theory and Methods, 32, 1317–1336. Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series (volumes 1, 2 and 3). Amsterdam: Gordon and Breach Science Publishers. Surles, J. G. and Padgett, W. J. (2005). Some properties of a scaled Burr type X distribution. Journal of Statistical Planning and Inference, 128, 271–280.

3

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.4,527-535,2008, COPYRIGHT 2008 EUDOXUS PRESS, 527 LLC

On the Rate of Approximation of Meyer-K¨ onig and Zeller Operators

Xiao-Ming Zeng Department of Mathematics, Xiamen University, Xiamen 361005, China E-mail: [email protected]

Abstract In this paper the asymptotic property of Meyer-K¨onig and Zeller operators Mn for bounded functions on [0, 1] is studied. An asymptotic convergence theorem of this type approximation is established by means of some probabilistic methods and results and accurate estimate technique to the basis functions of the operators Mn . The main result of this paper subsumes the approximation of the operators Mn for functions of bounded variation as a special case. 2000 Mathematics Subject Classification: 41A10, 41A36, 41A25. Keywords: Rate of approximation, Meyer-K¨onig and Zeller operators, Probability distribution, Basis functions.

1

INTRODUCTION

For a function f defined on [0, 1], Meyer-K¨onig and Zeller operator Mn [10] is defined by ∞ X k )mn,k (x), 0 ≤ x < 1, Mn (f, x) = f( n+k k=0 Ã

Mn (f, 1) = f (1),

mn,k (x) =

n+k−1 k

!

xk (1 − x)n .

(1)

If replacing the basis function mn,k (x) in definition (1) with the new basis à ! n+k function m ˆ n,k (x) = xk (1 − x)n+1 , one get a modified version of Mn , k which belongs to Cheney and Sharma [4]. The asymptotic convergence properties of Bernstein type operators for bounded functions have been studied in [11]. In 1

528

X-M ZENG

this paper we study the asymptotic convergence property of the operators Mn for bounded functions on [0, 1]. By means of some probabilistic methods and results and accurate estimate technique to the basis functions mn,k , we establish an asymptotic convergence theorem of this type approximation. Our investigation subsumes the approximation of operators Mn for functions of bounded variation as a special case. The following three quantities were first introduced in [11]. For their basic properties, one refers to [11]. Ωx− (f, δ1 ) =

sup

|f (t) − f (x)|,

Ωx+ (f, δ2 ) =

t∈[x−δ1 ,x]

Ω(x, f, λ) =

sup

sup

|f (t) − f (x)|,

t∈[x,x+δ2 ]

|f (t) − f (x)|,

t∈[x−x/λ,x+(1−x)/λ]

where f ∈ I, x ∈ [0, 1] is fixed, 0 ≤ δ1 ≤ x, 0 ≤ δ2 ≤ 1 − x, and λ ≥ 1. The following example shows that in the case of approximation of functions of bounded variation, the above quantities may give better asymptotic estimate than using the total variation of function of bounded variation. (

Example 1. Consider the function f0 (x) =

x2 sin(π/x), x ∈ (0, 1] . 0, x=0 0

f0 (x) is bounded variation on [0, 1] by the boundedness of f0 (x). On the interval [0, n−1 ] taking points: 1 1 1 1 1 1 > > > > ... > > > 0. n n + 1/2 n+1 (n + 1) + 1/2 n+n (n + n) + 1/2 It is easy to observe that µ

1/n

_ 0

(f0 ) ≥

1 n + 1/2

¶2

µ

+

1 (n + 1) + 1/2 µ

1 > (n + 1) 2n + 1/2

¶2

¶2

µ

+ ... +

1 (n + n) + 1/2

¶2

> (4n)−1 ,

and obviously, Ω0+ (f0 , n−1 ) ≤ n−2 . The main result of this paper is as follows: Theorem 1. Let f be bounded on [0, 1], f (x+) and f (x−) exsit at a fixed point x ∈ (0, 1) and r = x/(1 − x). Then for n ≥ 2 we have ¯ ¯ n √ X ¯ Af,n,x ¯¯ 4 ¯Mn (f, x) − f (x+) + f (x−) − √ ≤ Ω(x, gx , k) + O(n−1 ), ¯ ¯ 2 nx(1 − x) k=1 3 2πxn (2) 2

RATE OF APPROXIMATION...

529

where gx (t) is defined by    f (t) − f (x+), x < t ≤ 1;

gx (t) =

0,

t = x;

(3)

  f (t) − f (x−), 0 ≤ t < x.

and (

Af,n,x =

3(1 − x)f (x) + (2x − 1)f (x+) + (x − 2)f (x−), nr = [nr] , (2x − 1 + 3(1 − x)(nr − [nr]))(f (x+) − f (x−)), nr = 6 [nr]

(4)

in (4), [nr] denotes the greatest integer not exceeding nr. From Theorem 1 we get an interesting asymptotic formula as follows. Corollary 1. Under the conditions of Theorem 1, if Ω(x, gx , λ) = o(λ−1 ), then we have the following asymptotic formula Mn (f, x) =

Af,n,x f (x+) + f (x−) + o(n−1/2 ). + √ 2 3 2πxn

(5)

We point out that approximation of functions of bounded variation is the spacial case of Theorem 1. From Theorem 1 we get immediately W

Corollary 2. Let f be a function of bounded variation on [0, 1], and let ba (f ) denote the total variation of f on [a, b], x ∈ (0, 1) and r = x/(1 − x). Then for n ≥ 2 we have ¯ ¯ Af,n,x ¯Mn (f, x) − f (x+) + f (x−) − √ ¯ 2

¯ n √ X ¯ 4 ¯≤ Ω(x, gx , k) + O(n−1 ) ¯ nx(1 − x) k=1 3 2πxn √

≤

n x+x/ X _

4 nx(1 − x) k=1

√

k

(gx ) + O(n−1 ),

x−x/ k

where Af,n,x and gx (t) are defined as in Theorem 1.

2

A SET OF LEMMAS

Each of the following four lemmas will be required in the proof of Theorem 1.

3

(6)

530

X-M ZENG

Lemma 1. For n ≥ 2, x ∈ [0, 1] there holds Mn ((t − x)2 , x) ≤

2x(1 − x) . n

(7)

P roof . By [2, Lemma 2.1] and simple calculation, for n ≥ 2 there holds ∆=

∞ µ X k=0

¶2

k −x n+k

µ

(n + k)! k 2x x (1 − x)n+1 ≤ 1 + k!n! n−1

Thus Mn ((t − x)2 , x) ≤

¶

x(1 − x)2 . n+1

∆ 2x(1 − x) ≤ . 1−x n

Using Bojanic-Cheng-Khan’s method [3, 5, 9] and Lemma 1 we obtain Lemma 2. For gx (t) defined in (3) we have n √ X 4 Ω(x, gx , k). nx(1 − x) k=1

|Mn (gx , x)| ≤

(8)

Because the method of proof of Lemma 2 is well known (cf. [3, 5, 7, 9, 12]), we here omit the details of the proof. Lemma 3. Let {ξk }∞ k=1 be a sequence of independent and identically distributed random variables with the expectation E(ξ1 ) = a1 , the variance E(ξ1 − a1 )2 = σ 2 > 0, E(ξ1 − a1 )4 < ∞, and let Fn stand for the distribution function of n √ P (ξk − a1 )/σ n. If Fn is a lattice distribution and Fn∗ is a polygonal approximak=1

tion of Fn (see the following Definition 1), then the following equation holds for all t ∈ (−∞, +∞) 1 Fn∗ (t) − √ 2π

Z

t

−∞

e−u

2 /2

du −

E(ξ1 − a1 )3 1 2 √ (1 − t2 ) √ e−t /2 = O(n−1 ). 3 6σ n 2π

(9)

The proof of Lemma 3 can be found in [6, p. 540-542]. Definition 1 ([6, p. 540, Definition]). Let F be concentrated on the lattice of points b ± nh, but on no sublattice (that is, h is the span of F). A polygonal approximation F ∗ to F is a distribution function with a polygonal graph with vertices at the midpoints b ± (n + 1/2)h lying on the graph of F. Thus F ∗ (t) = F (t)

if t = b ± (n + 1/2)h; 4

(10)

RATE OF APPROXIMATION...

F ∗ (t) = 1/2[F (t) + F (t−)]

531

if t = b ± nh.

(11)

The following Lamma 4 is an accurate estimate technique to the basis functions of the operators Mn , which is a key auxiliary result in the proof of Theorem 1. Lemma 4. For x ∈ (0, 1), r = x/(1 − x), we have

and

³ ´ 1−x mn,[nr] (x) = √ + O (nx)−3/2 , 2πxn

(12)

³ ´ 1−x + O (nx)−3/2 . mn,[nr]+1 (x) = √ 2πxn

(13)

P roof . We first show that µ

nr [nr]

¶[nr]+1/2 µ

³

Set W1 (n, r) = (0 ≤ ε < 1), then

nr [nr]

n + [nr] n + nr

´[nr]+1/2 ³

µ

W1 (n, r) = 1 +

ε [nr]

¶n+[nr]−1/2

= 1 + O([nr]−1 ).

´ n+[nr] n+[nr]−1/2 , n+nr

¶[nr]+1/2 µ

1+

ε n + [nr]

(14)

and write nr = [nr] + ε ¶−(n+[nr]−1/2)

.

Thus µ

logW1 (n, r) = ([nr] + 1/2)log 1 + Ã

µ

ε ε = ([nr]+1/2) +O [nr] [nr]

¶2 !

ε [nr]

¶

µ

− ([n + [nr] − 1/2)log 1 + Ã

µ

ε n + [nr]

ε ε −([n+[nr]−1/2) +O n + [nr] n + [nr]

= O([nr]−1 ), which implies that W1 (n, r) = 1 + O([nr]−1 ). Using Stirling’s formula: √ n! = (n/e)n 2πneθn /12n , 0 < θn < 1,

5

¶

¶2 !

532

X-M ZENG

and by direct calculations we find that √ √ 2πnx 2πnx (n + [nr] − 1)! [nr] mn,[nr] (x) = x (1 − x)n 1−x 1 − x [nr]!(n − 1)! √ 2πnx n (n + [nr])! [nr] = x (1 − x)n 1 − x n + [nr] [nr]!n! =

(n + [nr])n+[nr]−1/2 [nr]+1/2 x (1 − x)n−1 ec(x,n) , [nr][nr]+1/2 nn−1 µ

= where −

nr [nr]

¶[nr]+1/2 µ

n + [nr] n + nr

¶n+[nr]−1/2

ec(x,n) ,

1 1 1 − ≤ c(x, n) ≤ . 12n 12[nr] 12n

Thus, it follows from Eq. (12) that √ 2πnx m (x) = 1 + O([nr]−1 ), 1 − x n,[nr] which derives the estimation (14). Furthermore, note that µ

mn,[nr]+1 (x) − mn,[nr] (x) = mn,[nr] (x)

¶

n + [nr] x−1 , [nr] + 1

and since nr = [nr] + ε (0 ≤ ε < 1), then n + [nr] (1 − x)([nr] + ε) + [nr]x − [nr] − 1 ε − εx − 1 x−1= = , [nr] + 1 [nr] + 1 [nr] + 1 that is

µ

¶

ε − εx − 1 mn,[nr]+1 (x) = mn,[nr] (x) +1 . [nr] + 1 Thus, we get (13) directly from (12). The proof of Lemma 4 is completed.

3

PROOF OF MAIN RESULT

Proof of Theorem 1. For any f ∈ IB , if f (x+) and f (x−) exist at x, by BojanicCheng decomposition it follows that Mn (f, x) −

f (x+) + f (x−) f (x+) − f (x−) = Mn (sgnx , x) 2 2 6

RATE OF APPROXIMATION...

+

533

2f (x) − f (x+) − f (x−) Mn (δx , x) + Mn (gx , x). 2

(15)

   1,

( t>x 1, t = x 0, t = x , and δx (t) = where gx (t) is defined in (3), sgnx (t) = .  0, t 6= x  −1, t < x

We need to estimate every term in the right side of (15). The term Mn (gx , x) has been estimated in Lemma 2. Let r = x/(1 − x). Direct calculation gives (

Mn (δx , x) =

mn,[nr] (x), nr = [nr] . 0, nr = 6 [nr]

(16)

Below we estimate Mn (sgnx , x). Let {ξi }∞ i=1 be a sequence of independent random variables with the same geometric distribution P (ξi = k) = xk (1 − x), k = 0, 1, 2, · · ·, and x ∈ (0, 1) is a parameter. Direct computations give x x , E(ξ1 − Eξ1 )2 = , 1−x (1 − x)2

Eξ1 =

x2 + x x3 + 7x2 + x , E(ξ1 − Eξ1 )4 = < ∞. 3 (1 − x) (1 − x)4

E(ξ1 − Eξ1 )3 = Let ηn =

n P i=1

ξi and Fn stand for the distribution function of

Then the probability distribution of the random variable ηn is Ã

P (ηn = k) =

n P i=1

√ (ξi − Eξ1 )/σ n.

!

n+k−1 k

xk (1 − x)n = mn,k (x).

Thus X

Mn (sgnx , x) = −

knx

mn,k (x) −

k