Weighted Approximation with Varying Weight (Lecture Notes in Mathematics, 1569) 354057705X, 9783540577058

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen

1569

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen

1569

Vilmos Totik

Weighted Approximation with Varying Weight

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Author Vilmos Totik Bolyai Institute University of Szeged Aradi v. tere 1 6720 Szeged, Hungary

and Department of Mathematics University of South Florida Tampa, FL 33620, USA

Mathematics Subject Classification (l991): 41AlO, 41A17, 41A25, 26Cxx, 31AlO, 31A99, 41A21, 41A44, 42C05, 45E05

ISBN 3-540-57705-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57705-X Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging-in-Publication Data. Totik, V. Weighted approximation with varying weights /Vilmos Totik. p. em, - (Lecture notes in mathematics; 1569) Includes bibliographical references and index. ISBN 3-540-57705-X (Berlin: softcover: acid-free). - ISBN 0-387-57705-X (New York: acid-free) 1. Approximation theory. 2. Polynomials. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1569. QA3.L28 no. 1569 [QA221] 510 sdc20 [511'.42] 93-49416 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany SPIN: 10078788

46/3140-543210 - Printed on acid-free paper

Contents 1 Introduction

1

I

7

Freud weights

2

Short proof for the approximation problem for Freud weights

3

Strong asymptotics 3.1 The upper estimate. 3.2 The lower estimate 3.3 The V case . . . . .

II 4

Approximation with general weights A general approximation theorem 4.1 Statement of the main results . 4.2 Examples and historical notes.

7 10 14 17 19

21 21

21

23

5 Preliminaries to the proofs

25

6

Proof of Theorems 4.1,4.2 and 4.3

32

7 Construction of Examples 4.5 and 4.6 7.1 Example 4.5 . 7.2 Example 4.6 .

38

III

49

Varying weights

38

44

8

Uniform approximation by weighted polynomials with varying weights 49

9

Modification of the method The lower estimate . . . . The upper estimate. . . . The asymptotic estimate .

57

59 60 64

10 Approximation in geometric means

70

IV

79

Applications

11 Fast decreasing polynomials

79

12 Approximation by W(anx)Pn(X)

85

VI

13 Extremal problems with varying weights

91

14 Asymptotic properties of orthogonal polynomials with varying weights 95 15 Freud weights revisited

103

16 Multipoint Pade approximation

106

17 Concluding remarks

109

References

111

Index

115

1

Introduction

In this work we are going to discuss polynomial approximation with weighted polynomials of the form w n Pn , where w is some fixed weight and the degree of Pn is at most n. We emphasize that the exponent of the weight w n changes with n, so this is a different (and in some sense more difficult) type of approximation than what is usually called weighted approximation. In fact, in the present case the polynomial Pn must balance exponential oscillations in ui", To have a basis for discussion let us consider first an important special case. Let w(x) = exp(-clxIO"), c> 0 be a so called Freud weight. H. N. Mhaskar and E. B. Saff [34] considered weighted polynomials of the form w n Pn , where the degree of Pn is at most n. They found that the norm of these weighted polynomials live on a compact set Sw, i.e. for every such weighted polynomial we have IIwnpnllR = II wnpnlls"" furthermore, w n Pn/llw n Pnlls", tends to zero outside Sw. They also explicitly determined Sw: (1.1)

where

s; =

1 1

10":=

0

vO"-l

v'f=V2 dv

0: 1 0: = f(2')f(2)/(2f(2'

1

+ 2))'

(see Section 3 below). One of the most challenging problems of the eighties in the theory of orthogonal polynomials was Freud's conjecture (see Section 3) about the asymptotic behavior of the recurrence coefficients for orthogonal polynomials with respect to the weights w. The solution came in three papers [16], [29] and [27] by D. S. Lubinsky, A. Knopfmacher, P. Nevai, S. N. Mhaskar and E. B. Saff. The most difficult part of the proof was the following approximation theorem ([29]). Theorem 1.1 If wa(t) = exp( -'0" Itla), 0: > 1 is a Freud weight normalized so that Sw", = [-1,1], then for every continuous f which vanishes outside (-1,1) there are polynomials Pn of degree at most n, n = 1,2, ... such that uniformly tends to f on the whole real line.

Let us mention that it follows from what we have said about tending then it to zero outside [-1, 1], if f can be uniformly approximated by must vanish outside [-1,1]. In the next section we shall present a rather elementary and direct proof for Theorem 1.1. Then, in Section 3, we shall derive a short proof for the strong asymptotic result of Lubinsky and Saff for an extremal problem associated with Freud weights. With this we will provide a self contained and short proof for the most important result of the monograph [28]. In Section 4.1 we shall considerably generalize Theorem 1.1 and solve the analogous approximation problem for a large family of weights. In earlier works the approximation problem was mostly considered for concrete weights such as

2

Introduction

Section 1

Freud, Jacobi or Laguerre weights. The generalization given in Theorem 4.2 is the first general result in the subject and is far stronger than the presently existing results (e.g. it allows Sw to lie on different intervals). It also solves several open conjectures. However, the new and relatively simple method is perhaps the most important contribution of the present paper (Lubinsky and Saff themselves generalized Theorem 1.1 in a different direction, see e.g. [28] and Section 12). We shall first restrict our attention to the important special case given in Theorem 1.1 in order to get a simple proof for the above mentioned asymptotics (and hence for the so called Freud conjecture) and in order not to complicate our method with the technical details that are needed in the proof of Theorem 4.2 (see Section 5). In the third part of this work we shall present a modification of the method. This will allow us to consider varying weights in the stronger sense, that we shall allow even W n to vary with n. Recently a lot of attention has been paid to such varying weights which are connected to some interesting applications to be discussed in Chapter IV. In essence our approximation problem can be reformulated as follows: how well can we discretize logarithmic potentials, i.e. replace them by a potential of a discrete measure which are the sums of n (n = 1,2, ...) equal point masses (see the discussion below for the relevant concepts). The usual procedure is the following: divide the support into n + 1 equal parts with respect to the measure and place masses l/n at these division points. This approach has proven to be sufficient and useful in many problem. However, the process introduces singularities on the support which has to be avoided in finer problems. Our method in its simplest form is a modification of the previous idea. We also divide the support into n equal parts with respect to the measure, but we use the weight points of these parts instead of their endpoints for placing the mass points to, then we vertically shift this discrete measure by an amount Ln/n where L n - 00 is appropriately chosen. This modification will result in a dramatic increase in the speed of approximation.

*************

In the rest of this introduction we shall briefly outline the results from the theory of weighted potentials that we will need in the paper. We shall use logarithmic potentials of Borel measures. If J.l is a finite Borel measure with compact support, then its logarithmic potential is defined as its convolution with the logarithmic kernel:

UI'(z) =

J

log Iz

tl dJ.l(t).

Let E be a closed subset of the real line. For simplicity we shall assume that "E is regular with respect to Dirichlet problem in C \ R", by which we mean that every point 3:0 of E satisfies Wiener's condition: if

3

Introduction

Section 1

then

L logljcap(E n 00

(1.2)

- 00

n) -

n=l

(see the following discussion for the definition of the logarithmic capacity). In particular, this is true if E consists of finitely many (finite or infinite) intervals. This regularity condition is not too essential in our considerations, but it simplifies some of our proofs. A weight function w on E is said to be admissible if it satisfies the following three conditions (1.3)

(i)

w is continuous;

(ii)

E o := {x E Elw(x)

(iii)

if E is unbounded, then Ixlw(x) - 0 as

> O} has positive capacity;

Ixl- 00,

x E E.

We are interested in approximation of continuous functions by weighted polynomials of the form w n Pn . To understand the behavior of such polynomials we have to recall a few facts from [34] and [35] about the solution of an extremal problem in the presence of a weight (often called external field). We define Q = Qw by

w(x) =: exp(-Q(x)).

(1.4)

Then Q : E - (-00,00] is continuous everywhere where w is positive, that is where Q is finite. Let M(E) be the set of all positive unit Borel measures j.L with supp(j.L) E, and define the weighted energy integral

JJlog[lz - tlw(z)w(t)t1dj.L(z)dj.L(t) JJ [log Iz tl + Q(z) + Q(t)] dj.L(z)dj.L(t).

(1.5)

The classical case corresponds to choosing E to be compact and w == 1 on E: If j.L is a Borel measures with compact support on R, then its logarithmic energy is defined as

I(j.L) :=

J

U/S(z)dj.L(z) =

JflOg

Iz

tl dj.L(t)dJ.L(z).

If K is a compact set, then its logarithmic capacity cap( K) is defined by the formula (1.6) := inf {I(j.L) 1j.L E M(K)}. log Now the capacity of an arbitrary Borel set B is defined as the supremum of the capacities of compact subsets of B, and a property is said to hold quasieverywhere on a set A if it holds at every point of A with the exception of points of a set of capacity zero.

4

Introduction

Section 1

The equilibrium measure (see [51] or [17]) WK of K is the unique probability measure WK minimizing the energy integrals in (1.6). Its potential has the following properties: (1.7) (1.8)

for

UWK(Z) = log

1 cap(K)

zEC,

for quasi-every

z E K.

If K is regular (which means that its complement C \ K is regular with respect

to the Dirichlet problem), then we have equality for every

z

in (1.8).

Returning to the general case of weighted energies, the next theorem was essentially proved in [34] and [35]. Theorem A Let w be an admissible weight on the set E, and let

Vw := inf{Iw(J.t) IJ.t E M(E)}.

(1.9)

Then the following properties are true. (a) Vw isfinite. (b) There exists a unique J.tw E M(E) such that

Iw(J.tw) = Vw. Moreover, J.tw has finite logarithmic energy. (c) Sw := supp(J.tw) is compact, is contained in Eo (c.f. property (ii) above), and has positive capacity. (d) The inequality

UJiw(z) 2 -Q(z) + Vw -

J

QdJ.tw =: -Q(z) + r;

holds on E. (e) The inequality

UJiw(z)

-Q(z)

+ Fw

holds for all z E Sw. (f) In particular, for every z E Sw,

The proof is an adaptation of the classical Frostman method. In fact, in [34] and [35] property (d) was proved to hold for quasi-every z E E. But the regularity of E implies that then the set of points where

5

Introduction

Section 1

holds is dense at every point of E in the fine topology (see [12, Chapter 10] or [17, Chapter III]), hence the inequality in question is true at every z E E by the continuity of Q (where it is finite) and the continuity of logarithmic potentials in the fine topology. The measure J.Lw is called the equilibrium or extremal measure associated with w. Above we have used the abbreviation

r;

:= Vw -

J

QdJ.Lw

for this important quantity. We cite another theorem ofR. N. Mhaskar and E. B. Saff [34, Theorem 2.1]' which says that the supremum norm of weighted polynomials w n Pn lives on Sw. Let us agree that whenever we write Pn , then it is understood that the degree of Pn is at most n.

Theorem B Let w be an admissible weight on E of degree at most nand

R. If Pn is a polynomial

(1.10) then for all z E C

(1.11) Furthermore, (1.10) implies

(1.12)

Iw(zt Pn(z)1

M

for

z E E.

This theorem asserts that every weighted polynomial must assume its maximum modulus on Sw. Soon we shall see that Sw is the smallest set with this property. Theorem B is an immediate consequence of the principle of domination (see the proof of Lemma 5.1 in Section 5).

Part I

Freud weights In the first part of the paper we shall consider exponential type (also called Freud) weights. We shall illustrate our method on them. The other purpose of this part is to give a self-contained and relatively short proof for the strong asymptotic results of Lubinsky and Saff [28].

2

Short proof for the approximation problem for Freud weights

In this section we give a short and simple proof for Theorem 1.1. Let Q(x) = 'Yalxla, so that wa(x) = w(x) = exp(-Q(x)). First we simplify the problem. I. Obviously, it is enough to consider !,s that are positive in (-1,1) and less than, say, 1. Furthermore, we know that it is sufficient to approximate on, say, [-2,2], because w n Pn tends to zero outside [-3/2,3/2] (see Theorems A and B from the introduction and the formula (3.7) in Section 3). II. It is enough to approximate by the absolute values of weighted polynomials. In fact, if wnlPnl uniformly tends to VI, then w 2n lPn!2 uniformly tends to I, and here IPn 12 is already a real polynomial. This shows our claim when the degree n is even. For odd degree one can get the statement by approximating I/w with even degree polynomials and then by multiplying through by w. III. It is enough to show the following: for every e > 0 and L > 0 there is a continuous function gL and for every large n polynomials Qn of degree at most n such that with J f := [-1 + e, 1- ]

(2.1) where the remainder term RL(x) satisfies IRL(x)1 Gf / L uniformly for x E J« with some C, 1 independent of L, and for every x E [-3,3] (2.2) where D = DL'f is a constant independent of n. In fact, suppose this is true, and apply it to w A instead of w with some A > 1. The corresponding extremal support is [-OA' OA] with OA = A-l/a tending to 1 together with A, hence, by choosing A > 1 close to 1 and then applying the statement above to a smaller e if necessary, we can see that there are polynomials Q[n/Aj of degree at most [n/A] such that with some gL and RL as above

8

Section 2

Freud weights

and

D1n3 ,

wn(x)IQ[n/>.j(x)1

x E [-2,2].

Since 0 n - A[n/A] A, and the family of function {gL - sQ I 0 s I} (considered on [-1 + e, 1- f]) is compact, for every large n there are polynomials Sn-[n/>.j of degree at most n - [n/A] such that ISn-[n/>.j(x) - I(x) exp( -gL(x) + (n - A[n/A])Q(X))l

exp( -gL(x) + (n - A[n/A])Q(x))/ L, ISn-[n/>.j(x)1

and (2.3)

x E he,

l(x)exp(-gL(x) + (n - A[n/A])Q(x)),

n-4,

ISn-[n/>.j(x)1

x E he \ J e,

x E [-2,2] \ i..

Now we set Pn = Q[n/>.jSn-[n/>.j, which has degree at most n. If", > 0 is given, then choose first e > 0 so that the maximum of I outside he is smaller than "" then chooose A > 1 as above, and finally choose L large enough to have GelL < ",. Then our estimates show that for sufficiently large n the difference IwnlPnl- II is at most 3", on [-2,2], and this is what we need to prove. IV. Thus, we only have to verify (2.1) and (2.2). Let us consider the so called Ullman distribution JJw given by its density function l ar 1 u (2.4) v(t) = duo 2 - t2 1r ItJ

al vu

It is well-known (see the computation in Section 3, especially (3.6) and (3.7)) that w(x) and exp(UIJ.. (x)) differ on [-1,1] only in a multiplicative constant, and elsewehere the weight w(x) is smaller than exp( U IJ.. (x)) times this constant. Hence it is enough to show (2.1) and (2.2) with w = War replaced by exp(UIJ.. ). In doing so we are going to use the standard discretization technique for logarithmic potentials (d. [42] and [28]) with some modifications, but exactly these modifications permit good approximation. Let v be the density of the Ullman distribution JJw (see (2.4)), and let us to < t1 < ... < t« 1 into n intervals Ij, divide [-1,1] by the points -1 j 0,1, ... , n - 1 with JJw(Ij) l/n. Since v is continuous and positive in (-1,1), there are two constants c, G (depending on i) such that if Ij n J e2 1= 0, then c/n IIj I G In. Let

= =

=

=

s

ej:= (II.) JJ

J

f

}1;

t dJJ(t) = n

f

}1;

t dJJ(t)

be the weight point of the restriction of JJw to Ij, and set

Qn(t) = II(t - iL/n - ej). j

We claim that this choice will satisfy (2.1) and (2.2) (with w replaced by exp(UIJ.. )).

Section 2

Short proof for the approximation problem for Freud weights

9

First of all let us consider the partial derivative of UJ'w (z) at z = x + iy with respect to y:

1 1

auJ'w(z)

ay

(2.5)

=-

-1

(

y X -

t

)2

+ y2v(t)dt

1rv(x)

as y 0 - 0 uniformly for x E J, by the properties of the Poisson kernel. This, and the mean value theorem implies that

. uJ'w(x) - uJ'w(x - zL/n) = 1rLv(x) n

(2.6)

+ 0 (L) ;

uniformly in x E J f • The same argument shows that (2.7) uniformly for x E R. Actually, (2.5) and (2.6) uniformly hold on R because v is continuous (even at ±1) and vanishes outside [-1,1]. We shall use this fact in Section 3, but for the present purposes we keep the above formulation because in Section 4.1 we shall consider weights the density of which is not necessarily continuous around the endpoints, and it will be easier to point out the necessary changes if we work with (2.6) and (2.7). Let J-ln(t) = J-lw(t - iL/n), i.e, we are defining J-ln on the interval [-1,1] + iL/n, which is obtained by shifting [-1,1] upwards on the plane by the amount L[n: Then the preceding two estimates tell us how far apart the two potentials UJ'w and UJ'" can be on [-1,1] and on R. Next we estimate for x E Jfl x E Ijo

(2.8)

L n 1(log Ix - iL/n - tl-Iog Ix - iL/n - ej I) dlJw(t)

n-1 j=O

1;

Here the integrand is

I

e.- t e I n-j

(1+ x-ze· -nt -e' ) j

log 1+ x-z

Since the absolute value of

x - iL/n -

ej'

t E Ij

is at most 1/2 for large L (check this separately for lej - tl :::; Cln and for the opposite case which can only occur if Ij n = 0 and hence Ix - ej I > f/2 while IIj I < (2), it easily follows that then the last expression can be written in the form

10

Section 3

Freud weights

and since the integral of the first term on lj against dpw(t) is zero because of the choice of we have to deal only with the second term. For it we have the upper estimate

ej,

o(

(L/n)2

(G/n)2

+ (c(j - io)/n)2

)

oC;P) otherwise (recall that x E J f ) , hence we can continue (2.8) as

if n is sufficiently large. Now

and here, by the preceding estimate, the first term is at most G f / L in absolute value, while by (2.6) the second term is 1rv(x)L + o(L) uniformly in x E J f as n - 00 . This gives (2.1) (recall that we are working with exp(Ul'w) instead of

w).

The proof of (2.2) is standard: using the monotonicity of the logarithmic function we have for example for x E ljo' io < i < n - 1 the inequality log Ix - iL/n -

ej I

n

f

i.:

log Ix - iL/n - tldpw(t),

and adding these and the analogous inequalities for easily deduce the estimate

(2.9)

+

L

jo+l j=jo-l

log IQn(x)1

+ nUl'n(x)

n

-'L/

J

log

one can

3 log 6 +

1

1

x

z n

t

for every x E [-3,3]. This and (2.7) prove (2.2).

3

i < io together

I

Strong asymptotics

The theorems of this section are not new, they can be found in the monograph [28] by D. S. Lubinsky and E. B. Saff. We closely follow many steps from [28], but we substitute the approximation part of the proof with the simple method of Section 2 which allows us to make shortcuts and simplifications, thereby

Section :J

Strong asymptotics

11

significantly reducing the length of the original proof (which is scattered through about 100 pages). First we shall consider the L 2 extremal problem and then the LP one at the end of the section. In order to have a complete proof we add a few standard calculations that may help the reader. Let w(x) = wa(x) = e-"Yc>I:r:IC>, a> 1 be a Freud weight on R normalized so that Sw = [-1, 1] (this normalization is made for convenience, any other positive constant can replace "'fa on the right; for the explicit form of the constant "'fa see (3.5) below), and consider the orthonormal polynomials with respect to w 2 : Pn(W; x) = "'fn(w)x n + ...

defined by the orthogonality relation

J

Pn(w; X)Pm(Wj x)w

2(x)

dx = on,m.

When a = 2 these are the classical Hermite polynomials, for other a's G. Freud started to investigate their properties. Let lIn denote the set of polynomials of degree n and leading coefficients 1, i.e. lIn = {e" + ...}. The leading coefficient "'fn (w) of the orthonormal polynomials Pn are closely related to a weighted extremal (minimum) problem, namely

(3.1)

-1( )2 = "'fn W

. f In

PnEII n

J

pn2W 2 ,

and it is one of the most important quantities related to P«- In fact, their behavior determines the behavior of the Pn's which can also be seen by the fact, that in the recurrence formula xPn(W; x) = An+1Pn+I(Wj e ) + AnPn-l(Wj x)

the recurrence coefficients are given by

In [6] G. Freud made two conjectures: one on the asymptotics of the largest zeros of the Pn's and another one on the recurrence coefficients. E. A Rahmanov [42] solved the first conjecture, but the second one, which claimed that

(3.2)

lim n- 1 / aAn

n--+oo

=2

was open for some while, until it was settled in a series of papers [16], [29] and [27] by D. S. Lubinsky, A. Knopfmacher, P. Nevai, S. N. Mhaskar and E. B. Saff

12

Section 3

Freud weight.

(Freud himself verified the conjecture for a = 2,4,6, and for even integers it was settled by A. Magnus [32]). This was a typical conjecture that was obviously bound to be true (already Freud new that the terms on the left are in between two positive constants, and if the limit exists then it has to be 1/2; and there was no reason why the limit should not exist), but its proof required genuinely new tools. Shortly after settling Freud's conjecture, D. S. Lubinsky and E. B. Saff [28] proved the following strong (as opposed to (3.2), which is called ratio) asymptotics for the 'Yn(w)'s themselves, which is probably one of the all-time best results in the theory of orthogonal polynomials: lim 'Yn(W)1r1/22-ne-n/Otn(n+1/2)/0t = 1.

(3.3)

n .....Hxl

Below we shall present a relatively short proof for (3.3) that utilizes the approximation technique in Section 2. The original proof is scattered through the monograph [28] and is quite long. Let us start with the Ullman distribution given by its density

(3.4)

v(t)

= -a

1 v'u 1

1r

u

ItI

Ot- 1 2

-

du

t2

on [-1, 1]. For its potential we have (writing instead of the measure its density as a parameter in U) by switching the order of integration

11

- UV ( x ) =

o

au Ot_11jU1oglx-tldd t u. 2 1r

-u

';u 2 - t

The expression after u Ot- 1 is nothing else than the negative of the equilibrium potential of the interval [-u, u], hence it equals log u - log 2 if Ixl u and log Ix + v'x 2 - u 2 1- log 2 if IxI > u. Thus, for -1 x 1 we can continue the above equality as -log 2 +

1

-log 2 -

1

[e]

au Ot-

+ Ixl

1log

u du +

1 1

Ot

r

au Ot-

1log

0

Ix + J x 2 -

dV) .

av Ot- 1log(1 +

Integration by parts yields that the last integral is

=

1 1 o

vOt-l

v2

1 1

-

(3.5)

'YOt:=

t'

Jo

v Ot- 1

we finally get for x E [-1,1]

v Ot- 1

1 a'

0

+

where we used the identity v 2 = (1-

a

1

u 2 1 du

Since

a

1

Section 3

13

Strong asymptotics

1

(3.6) Let now Ixl > above

UV ( x ) = - ' Y " l x l " + l o g 2 + - . a 1, x E R. By symmetry we can assume x

U (x) = log 2 V

-1

1

au"-llog(x +

Jx

2 -

>

1. Exactly as

u 2)du,

and by differentiation we get

This tells us first of all that outside [-1,1] we have on R (3.7)

and so by (3.6) and Lemma 5.1 to be proven in Section 5 we can conclude that djtw(t) = v(t)dt and (3.8) Fw = log 2 + l/a. It also follows that for Ixl E (1,2)

(UV(x))' + a'Y"x"-l -llxl-

11 1/ 2 ,

where - indicates that the ratio of the two sides lies in between two absolute constants (in the range of the arguments indicated), and so (3.9)

for Ixl E (1,2), and (3.10)

(UJ'w(x) - Fw ) + 'Y"lxl" -Ixl"

when Ixl > 2. This and Theorem B of the introduction easily imply the followinginequality of D. S. Lubinsky [25]: for Pn = 1 + n- 7 / 12 (3.11)

= 1 + 0(1) as n -> 00 (for completeness we shall give a short proof for (3.11) at the end of this section). The awkward looking /r/" in the limits of integration on the left is just to match the proof below, we shall only need that it is larger than 1 + cn- 7 / 12 with some c > O. After these preliminaries let us return to (3.3). Let 00 we can conclude that F Fw . This argument can be repeated with j.tw and a interchanged, hence we get that F = Fw , and then that the two potentials UJ.lw and U I1 concide everywhere. But then j.tw = a (see e.g. [17, Theorem 1.12']. The last statement in the lemma follows from the same considerations if we note that sets consisting of finitely many points must have zero j.tw - and (f-measures because these measures have finite logarithmic energy.

I

Now we need the so called Fekete or Leja points (see [35]) associated with w. Let w be an admissible weight on the closed set E R. For an integer n 2 we set

The supremum defining 6;:' is obviously attained for some set

These F n are called n-th Fekete sets associated with w, or shortly w-Fekete sets. For fixed n, the sets :Fn need not be unique; however in our consideration below we can use any choice of them. We shall need that the asymptotic distribution of Fekete points is the same as the equilibrium distribution j.tw: if V:F

"

1 (A) := n

""

L.J

xef"nA

where A is any Borel subset of C (i.e. Vf" puts mass lin to every point Zj E F n ) , then (5.2) lim vr; = j.tw n-oc

in the weak" topology of measures (see [49, Lemma 2.2]). With this property at hand we can easily verify that Sw is the smallest compact set S with the property that every weighted polynomial attains its norm on S (c.f. Theorem B in the introduction). Lemma 5.2 Let w be an admissible weight on E, and let S E be a closed set. If, for every n = 1,2, ... and every polynomial Pn with deg Pn n,

then

s;

S.

27

Preliminaries to the proofs

Section 5

Proof. Consider the Fekete sets F n associated with w. We claim that for each n we can choose F n S. This will prove the lemma because then the normalized counting measures VF" associated with F n have support in S and converge in the weak" topology to the measure J.lw (see (5.2», hence Sw = supp(J.lw) S. Let F n = {tl, ... , tn} be any w-Fekete set, and suppose that tl, ... , t'-l E S but i, ... , t n tI. S. Consider the polynomial Pn-1(z) :=

II (z -

tk)'

k#,

By the choice of w-Fekete points maxIPn_1(z)!w(zt- 1 = IPn_1(t,)!w(t,t- 1. zEIJ

By our assumption the weighted polynomial w n - 1 Pn modulus somewhere on S, hence there is a E S with

t;

1

attains its maximum

max IPn_1(z)lw(zt- 1 = IPn_1(t;)lw(t;t- 1. zEIJ

Thus, together with {tl,oo.,t'_l,t"t'+l,oo.,t n}, the set {tl,oo.,t'_l,t;, t,+l,"" tn} will also be a w-Fekete set, and the latter set has already s points in S. We can continue this process and eventually arrive at a w-Fekete set contained in S.

I

Now we are ready to prove a characterization of the points in the support that will be useful in our further considerations. Lemma 5.3 Let w be an admissible weight on E. Then z E E belongs to the support Sw of the extremal measure J.lw if and only if for every neighborhood B of z there exists a weighted polynomial w n Pn, deg Pn n, such that wnlPnl takes its maximum on E in B n E, and nowhere else.

Proof. Let z E Sw and let B be any neighborhood of z. By applying Lemma 5.2 to the set S = E \ B, we get a weighted polynomial Pnw n with

IIwn PnIlIJ\B < IIwn PnIIIJ, which shows that w n IPn I takes its maximum on E in B n E, and nowhere else. Conversely, ifwnlPnl takes its maximum on E only in EnB, then by Theorem B (see the introduction) we must have B n Sw =f 0, and of course if this is true for every neighborhood B of z, then z E Sw.

I

As an immediate consequence we obtain Lemma 5.4 If oX> 1, then i.e. the sets

s;: s.,

decrease as oX increases.

28

Section 5

Approximation with general weights

Proof. Let Zo E and let B be any neighborhood of zoo By Lemma 5.3 there are a natural number n, a polynomial Pn of degree at most n, a point Zl E B n E and an TJ > 0 such that IwnA(zdPn(zdl = 1 + TJ, but outside B we have IwnAPnl :::; 1. Then w must be positive at Zl, say w(zd m > O. Furthermore, let M be an upper bound for w. We may assume m < 1 < M. For a positive integer I consider the polynomial We clearly have

(1 + TJ)l m ,

(w(x))[lnA]+lIPn(x)11 while for x E E \ B

(W(Zd)[lnA]+lIPn(zdl l :::; M.

For large I this means that the weighted polynomial

w[lnA]+l pi

n'

with deg :::; nl :::; [InA] + 1 takes its maximum modulus only in B. Since this is true for any neighborhood of zo, we can infer from Lemma 5.3 that Zo E Sw, as we have claimed.

I

Now we need the concept of balayage measure. Consider in C an open set G with compact boundary 8G, and let J1. be a measure with supp(J1.) G. The problem of balayage (or "sweeping out") consists of finding a new measure Ii supported on 8G such that 117i11 = 1IJ1.11 and (5.3)

Uf.I(z)

= UiT(z)

for quasi-every

z ¢ G.

For bounded G such a measure always exists ([17, Chapter IV, §2/2]) , but for unbounded ones (with cap(8G) > 0) we have to replace (5.3) by (5.4)

Uf.I(z) = UiT(z)

+c

for quasi-every

z ¢ G.

with some constant c. Besides (5.3)-(5.4) we also know that (5.5) respectively (5.6)

UiT(z)

s Uf.I(z) + c

holds for all z E C. Furthermore, if G is regular with respect to the Dirichlet problem, then we have equality in (5.3) and (5.4). This is the case for example when G is the complement of finitely many closed intervals on the real line. Very often we have to take the balayage of J1. out of G even if its support is not contained in G. In that case we take the balayage of the restriction of J1. to G onto 8G, and leave the part of J1. lying outside G unchanged. Now the first part of the following lemma is an immediate consequence of these properties, Theorem A and Lemma 5.1. The second part follows from Lemma 5.3.

Section 5

29

Preliminaries to the proofs

Lemma 5.5 Let w be admissible, and K Then

Sw a regular compact subset of Sw .

J.twiK =J.tw, where - indicates taking balayage onto K out ofC\K. In particular, SWI

K

= K.

The lemma is true without the regularity assumption but we will not need this stronger statement. Recall that K is called regular if C \ K is regular with respect to the Dirichlet problem. Let us also mention that regularity is characterized by the Wiener condition (1.2). The next lemma can be esasily proved by the method of Lemma 5.1 if we use the properties (1.7)-(1.8) of the equilibrium measures. Lemma 5.6 If.A

> 1 and J.tw

= Sw, furthermore this is a regular set, then

=

+

wsw·

Again, here the regularity can be dropped, but we shall anyway need the lemma only in the case when the support in question is an interval. In what follows, let vw denote the density function of J.tw (wherever it exists). If the two supports Sw and .A > 1 are not the same, then we only have inequalities for the corresponding extremal measures. Lemma 5.7 If.A

> 1, then J.tw

s

+

(1 -

and

Proof. The proof is based on the following theorem of de la Vallee Poussin from [5] (see also [4, 11, 7], and also [50]): Let J.l and v be two measures of compact support, and let n be a domain in which both potentials UIJ and U V are finite and satisfy with some constant c the inequality (5.7)

UIJ(z)

s UV(z) + c,

zEn.

s

If A is the subset ofn in which equality holds in (5.7), then vIA J'IA' i.e. for every Borel subset B of A the inequality v(B) J.t(B) holds. Consider the two potentials corresponding to the two measures J'w and J'w with some .A > 1. It follows from Theorem A (see the introduction) and (1.8) that with W = WSw (recall that this is the equilibrium measure of the set Sw)

30

Approximation with general weights

Section 5

for quasi-every z E Sw, hence by the principle of domination (see the proof of Lemma 5.1) we have this inequality everywhere (recall that if a measure /-I has finite logarithmic energy, then every set of zero capacity has zero /-I-measure). Furthermore, Theorem A,(f), (1.8) and Lemma 5.4 imply that the equality sign holds for quasi-every z E SwA. Now each of the measures /-Iw, /-IwA and w have finite logarithmic energy, hence they vanish on sets of zero capacity, thus by applying the above theorem of de la Vallee Poussin we can conclude the second inequality:

/-IwlSWA

±/-Iw A+

The first one can be shown with the same argument if we notice that with W = Ws A we have

.

for every z E SwA, hence by the principle of domination we have this inequality everywhere. Furthermore, by Theorem A,(f) and Lemma 5.4 the equality sign holds for quasi-every z E SwA. Thus, the first inequality follows as before from the aforementioned theorem of de la Vallee Poussin.

I

In order to apply the preceding lemma we shall need a convenient criterion for concluding that a point Xo from Sw belongs to some SWA, A > 1, as well (recall Lemma 5.4 that these support are decreasing). Lemma 5.8 Suppose Xo is a point in the interior of SWI V w is continuous in a neighborhood of Xo, and Vw (t) > (0 for It - Xo I (1 for some (0 > 0 and (1 > O. Then for A 1/(1 - (o(d the point Xo is in the interior of SWA, furthermore, VWA is also continuous in a neighborhood of xo.

Proof. We begin the proof by the following observation. If for w we consider minimizing the weighted energy (1.5) on E and on some closed set Sw E 1 E, then we arrive at the same extremal measure Sw. Seeing that SWA Sw (Lemma 5.4), this shows that in the proof we may assume without loss of generality that

E=Sw'

Let Vo be the measure the density of which is (0 on [xo - (1/2, Xo + (1/2] and consider the positive measure

o otherwise, and

of total mass 1, and the weight function

Section 5

31

Preliminaries to the proofs

that it generates. By Lemma 5.1 the extremal measure corresponding to W1 coincides with 111, and so Xo E SW1' Hence (see Lemma 5.3) if B is a neighborhood of Xo, th.en there is a polynomial Pn such that w? IPnl attains its maximum in B n E and nowhere else on E. The potential of the measure 1 1- fOf1

-:----110

is symmetric about Xo, attains its maximum at Xo and decreases to the right and increases to the left of Xo. But then for the weight

the weighted polynomial w? Pn can also attain its maximum only in B. Since this can be done for every neighborhood B of Xo, it follows that Xo E SW2' However, the weight function

and w>. with A = 1/(1 ­ fOfd differ on Sw only in a multiplicative constant, which, together with the relation E = Sw, means that J-lw x = J-lW2' and so = SW2' Thus, Xo E as is claimed in the lemma (see also Lemma 5.4). Furthermore, the same proof can be carried out with Xl in place of Xo for every Xl E [xo - fd2, Xo + fd2] lying sufficiently close to Xo, hence Xo is actually in the interior of and J-l)w>' has a continuous density in a neighborhood of Xo. It has left to show the continuity of the density function at Xo. Let I = [xo - fd2, Xo + fd2], and let ­ denote taking balayage onto lout of C \ I. By Lemmas 5.5 and 5.6 we have

and J-lwI I

=

+

(1­ ±)

WI,

from which we get the formula

I(R \ 1) =

AJ­lwI I ­ (A ­

AJ-lw II + AJ-lw I(R \ I)

(A ­ 1)WI ­

-

1)WI ­

I(R \ I)

I(R \ I).

Now on the right each term beginning with the second one has continuous density in the interior of I, furthermore, by our assumption the first term also has continuous density has continuous density at Xo, and these prove that function at Xo.

I

32

6

Section

Approximation with general weights

6

Proof of Theorems 4.1, 4.2 and 4.3

After the preliminaries of the preceding section we are in the position to prove Theorems 4.1-4.3.

Proof of Theorem 4.1. Let N 1 be the set of the even natural numbers. By considering (wn Pn)2 we can see that for n E N 1 there are polynomials Rn of degree at most n such that w n Rn converges to P on Sw U {xo}, and here P is already nonnegative. Let the minimum and maximum of P on Sw be m and M, respectively. Let us suppose on the contrary that a := P(xo) > 0, which will lead us to a contradiction as follows. We distinguish three cases.

°

Case I. P is not constant on Sw (i.e. m < M). We shall utilize the existence on [0, M] U{a}, of a polynomial U(x) without constant term such that U(x) U(m) i= U(M) and on [0, M] U {a} the polynomial U takes its maximum at the point a. Such U's can be constructed as follows. Let K = max{2M,a}. Ifwe choose the polynomial U*(x) as x(K - x)", then U* increases on the interval [0, KI(s + 1)] and decreases on [KI(s + 1), K], hence by selecting s bigger than Kf o: and setting U(x) = U*(Kxl(s+l)a), the polynomial U will be nonnegative on [0, K], will increase on [0, a] and decrease on [a, K]. If U(m) i= U(M), then we are ready. In the opposite case certainly m i= 0, and we can set U(x) = U**(x)U*(Kxl(s + l)a) with a polynomial U** that is nonnegative on [0, K], takes different values at m and M and takes its maximum on [0, K] at the point a. We claim that there is a subsequence N 2 N 1 and for each n E N 2 a polynomial Sn of degree at most n such that w n Sn uniformly tends to 9 := U(P) on Sw U {xo} as n -+ 00, n E N 2 • In fact, let k be the degree of U. Since w 2n R2n -+ n -+ 00, we can see that for any j k the weighted polynomials w 2k!n converge to (P)j· Thus, if U(x) = 2:::=1 ajx j, then the polynomials

°

r,

k

S2k!n(X) =

I: j=l

of degree at most 2k!n, will do the job (thus, we can choose N 2 = {2k!n I n = 1,2 ...}). By squaring again and considering g2 instead of 9 := U(P) if necessary, we can also suppose that the polynomials Sn are nonnegative. Furthermore, the degree of Sn can be exactly n, for we can always add factors of the form (L - x)2 I L 2 to Sn with sufficiently large L and these factors do not to change the properties of Sn on the compact set Sw U {xo}. In a similar manner, without loss of generality we may suppose that Sn does not vanish on the real line, for we can always replace any double real zero a of Sn by two conjugate complex zeros lying close to a. Note also that 9 is not constant on Sw because we had U(m) i= U(M) for the polynomial U.

33

Proof of Theorems 4.1, 4.2 and 4.3

Section 6

If we use Theorem A from the introduction we can see that

1

(6.1)

n(UJJw(x) - Fw ) -log Sn(x)

->

logg(x)

uniformly for all x E Sw U{xo} as n -> 00, n E .N2 . If M 1 denotes the maximum of g on Sw then it follows that there is a positive sequence {t"n} converging to osuch that

(6.2)

x E Sw·

Recall now the choice of U, according to which the function g attains its maximum M 1 at the point Xo on s; U {xo}. Thus, (6.3) as n -> 00, n E .N2 . Let t/« be the counting measure on the set of zeros of Sn (counting multiplicity) and let V n be its balayage out of C \ Sw onto Sw with balayage constant dn , i.e. (see (5.4) in Section 5) -

UVn(x)

1 = UVn(x) + dn = log Sn(x) + dn

for quasi-every x E Sw. Then (6.2) goes into - n(UJJw(x) - Fw )

(6.4)

+ Uii;'(x) - dn + logM1 + t"n

0

for quasi-every x E Sw, hence by the principle of domination (see the proof of Lemma 5.1) this inequality holds true for all x E C. On the other hand, from the nonincreasing character of taking balayage (see (5.6)) we can conclude from (6.3) that (6.5)

lim sup

n ....oo,

(-n(UJJw(xo) - Fw ) + Uii;'(xo) - dn + logM1 + t"n)

0,

hence we must have here equality in the limit. But the function on the left is harmonic and nonnegative on C \ Sw, so Harnack's theorem implies that

uniformly on compact subsets of C \ Sw. In particular I for (6.6)

lim

n ....oo,

(nFw

-

dn

z

=

00

we obtain

+ logMd = 0,

where we used that the total mass of the measures nJ.tw and V n are both n. (6.1) yields with some monotone decreasing sequence {7]n} tending to zero that

34

Section 6

Approximation with general weights

for quasi-every x E Sw, which combined with the preceding limit relation gives for n E N 2 Ml (6.7) - nUJjw(x) + UVn(x) log ( ) - Pn g X + TJn for quasi-every x E Sw, with some sequence {Pn} tending to zero. Since the equilibrium measure w = ws; of Sw has finite logarithmic energy, sets of zero capacity must have zero w-measure, hence the preceding inequality holds true w-almost everywhere. Thus, we can integrate this inequality with respect to w. Using the fact that the measures nJ1.w and V n have finite logarithmic energy (recall that Sn did not have real zero, hence the balayage measure of its zero counting measure is of finite logarithmic energy) and that we have

UW(x) = log

cap

w

for quasi-every x E Sw (see (1.8)) we get from Fubini's theorem that

J

nUJjwdw = n

J

UWdJ1.w = nlog

1 = cap(Sw)

J

UWdiJ;; =

J

U'ii;;dw,

and so during the integration of (6.7) against w the left hand side becomes zero. Thus, in the limit we get from the monoton convergence theorem that

J

u,

log g(x)dw(x) = 0,

which can happen only if g(x) = M l w-almost everywhere (recall that M l was the maximum of g on Sw). But this is certainly not the case for g is not constant on Sw, so in some neighborhood 0 of an Xl E Sw we have g < M l , and since cap(O n Sw) > 0 (recall that Sw was the support of the measure J1.w of finite logarithmic energy), we must have w(O n Sw) > O. The obtained contradiction proves the claim. Case II. P is constant on Sw, but w is not. This case can be easily reduced to the previous one, for if w 2n R 2n tends to a constant on Sw, then W 2n +2 R 2n will tend to w 2 which is not constant and deg R2n 2n + 2. Case III. Both P and ware constant on Sw (in this case J1.w = wsw), We can assume by proper normalization that w(x) == P (x) == 1 on Sw. By the maximum principle we have strict inequality in

UW(z) when

z

s log cap

w

Sw (see (1.7)), and since in the present case

Section 6

Proof of Theorems 4.1, 4.2 and 4.3

35

(see Theorem A,(d) in the introduction), we can conclude that w(xo) < 1. Hence by considering w2n+2R2n instead of w2nR 2n if necessary, we can assume that j2(xo) =: a =1= 1. Choose now a polynomial U without constant term so that U(I) < 1 < 3 < U(a) is satisfied. Then, as before, there are some polynomials Sn, n E./II2 such that wnSn -+ U(j2) uniformly on Sw U {xo}, i.e. Wn(x)Sn(x) uniformly converges to U(I) < 1 on Sw while it converges to U(a) > 3 at xo. This would mean that the sup norm of wn Sn on Sw is smaller than 2 for all large n E ./112 , and at the same time these weighted polynomials take values bigger than 2 at Xo - a contradiction to Theorem B in the introduction.

I

Proof of Theorem 4.2. Let f E Co(SW). We follow the proof of Theorem 1.1. In the present case J E denotes the set of points x E SW which are of distance e form the complement R \ SW. Let J* be an arbitrary finite interval. Eventually we will choose J* so that in E \ J* we have (6.8) UlJw(z) -Q(z) + Fw + 1

(d. Theorem A and the definition of the admissibility of w in the introduction), but for secure a free choice of J* later, it can be arbitrary at this point. First we show that it is enough to verify the following analogues of (2.1) and (2.2) for arbitrary J*: (6.9) where the remainder term RL(X) satisfies IRL(X)I with some C E 1, and uniformly in x E En J*

CEIL uniformly in x E J E

(6.10)

In fact, suppose this is true, and exactly as in Section 2 we apply it to w A instead of w with some A > 1. We can do this, because by Lemma 5.8 there is a A > 1 such that the set J E/ 2 is in the support of and it has continuous density there, furthermore, exactly as UlSw the potential is continuous everywhere; and these are the only properties that we shall use in deriving (6.9) and (6.10) below. Hence, by chosing A > 1 close to 1 we get that there are polynomials Q[n/A] of degree at most [nIA] such that with some 9L and RL as above

and (6.11)

xEEnr

where now J* is an interval satisfying (6.8). Let l/A < T < 1. We consider the polynomials Sn-[n/A] from the proof of Theorem 1.1, except that now we request that their degree be at most [(T -

36

Approximation with general weights

1f>.)n] (hence we write (2.3) only in the form

S[(T-l/A)n)

Section 6

for them), and we need their third property

(6.12) Since the disjoint sets J" \ J, and he consist of finitely many intervals, there is a 0 < e < 1 and for each m polynomials Rm of degree at most m such that (6.13)

IRm(x) - 11::; em

(6.14)

IRm(x)1

and (6.15)

s em

0::; Rm(x) s 1

for x E he, for x E J" \ J, for x E J e \ he

(see e.g. [15, Theorem 3] where such polynomials were constructed for two disjoint intervals, from which the Rm's with the stated properties can be easily patched together). Finally, we set

which has degree at most n. Exactly as in Section 2 we get that if 1] > 0, then by choosing e > 0, >. > 1 and L appropriately, for sufficiently large n the difference IwnIPn 1-II will be smaller than 31] on the set J" n l:. The only remark we have to make is that by (6.11), (6.12) and (6.14), the weighted polynomial w n Pn is exponentially small on (l: n r) \ J e . But by (6.8) and Theorem B (see the introduction) the same is true on l: \ J"; and this proves that

for every x E l: provided n is sufficiently large. Thus, it has left to prove (6.9) and (6.10). The proof of (6.9) is identical to the proof of (2.1) given in Section 2, there is no need to change anything. As for (6.10), with the measure J1.n from the proof of Theorem 1.1 we get by the continuity of the potential UJJw that

as n

--+ 00

uniformly in x E R, while the estimate

with D equal to the diameter of J" follows exactly as (2.9). These and

(see Theorem A,( d) in the introduction) prove (6.10), and the proof is complete.

I

Section 6

Proof of Theorems 4.1, 4.2 and 4.3

37

Proof of Theorem 4.3. We show that Theorem 4.3 is a consequence of Theorem 4.2. Let E = Ulj, where on the right we have a finite and disjoint union, and suppose that Q is convex on each of the Ij's. It was proved in [34, Theorem 2.2] that then Sw consists of finitely many intervals at most one lying in any of the

t,».

Let I be any subinterval of the interior of Sw. We recall Lemma 5.5 according to which

where indicates taking balayage onto lout of C \ I. The second measure on the right has continuous density inside I, and the same is true for the first one by [49, Lemma 4.5] (d. also the proof of Theorem 8.3) because the support of J.lw1 is I, and is a Cl+£-function on the support. Thus, we can conclude I 1 that J.lw has continuous density in the interior of Sw. In order to be able to apply Theorem 4.2 we have to show that the density V w of the extremal measure J.lw cannot vanish at any interior point of Sw. But this is an immediate consequence of Lemma 5.7 according to which we have

wl

and this clearly rules out that the density of J.lw vanishes at a point Xo unless Xo does not belong to the interior of Thus, if Xo E Sw does belong to the interior of some then at Xo the measure J.lw has positive density. But every interior point Xo of Sw must belong to the interior of at least one SwL In fact, since every consists of intervals at most one of which can lie in any Ij, it is enough to prove that in any neighborhood of any point Xo of Sw there is a point Xl lying in some because then this and the decreasing character of the supports (Lemma 5.4) imply our claim concerning every point in Int(Sw) lying in the interior of some But if Xo E Sw, and B is any neighborhood of Xo, then there are an n and a polynomial Pn of degree at most n such that wnlPnl attains its maximum in E at some point of BnE and nowhere outside of B. By continuity then the same is true of w>.nlPnl with some A> 1 sufficiently 1: 0. With this the proof is close to 1, and so Theorem B implies that B n complete.

s..

I

As for Theorem 4.4, compactness shows that it is enough to consider funcand for such functions the preceding tions that vanish outside some proof gives the appropriate approximation.

38

Approximation with general weights

7

Construction of Examples 4.5 and 4.6

Section 7

In this section we construct the two theoretically important Examples 4.5 and 4.6.

7.1

Example 4.5

In Example 4.5 we have to construct a weight w such that the support of the corresponding extremal measure is [-1, 1), this measure has continuous density in (-1,1) which is positive everywhere except at 0, and still no function that is nonzero at 0 is the uniform limit of weighted polynomials. Hence, in this case the largest set for the approximation problem of Section 4.1 is the restricted support (-I,O)U(O,I). The idea of the proof is to construct a density function v that is positive and continuous on (-1, 0) U (0, 1) and tends to zero very fast at the origin. We shall construct an even w by inductively choosing density functions that will converge in some sense to the density of w. In fact, for each k 2 we shall construct a function Vk with the following properties: Let Sk = [-1 + 2- k , -2- k) U [2- k, 1- 2- k). For every k 2 1. Vk is even, positive and continuous on Sk and zero elsewhere,

2. the integral of Vk is 1, i.e.

Vk generates an absolutely continuous positive measure of unit mass with support on Sk,

3. Vk(X)

Ijk if

Ixl

4. IVk(X) - Vk_l(X)1

2 1-

k

,

furthermore vk(±2-k)

Ij(k + 1),

2- k if x E Sk-l,

5. there exists an nk > k such that if Wk(X) = exp(Utlk(x)), where U'": denotes the potential associated with the measure generated by Vk, then for every polynomial Pn k of degree at most nk the inequality

implies

From here the construction of w with the desired properties will be simple, so first let us consider the construction of Vk. For k = 1 define Vk to satisfy the first three conditions, and now let us proceed with the construction of Vk+! provided Vk is already known. Let

Section 7

Construction of Examples 4.5 and 4.6

39

Note that Ik C Sk+l is disjoint from Sk, hence if WS k is the equilibrium measure associated with Sk (see Section 4.1), then (7.1)

(see (1.7)-(1.8) and apply the maximum principle according to which the potential of W s; is strictly smaller than 10g(1 I cap( Sk)) outside Sk)' Choose now a {3 = (3k > 1 arbitrarily. We claim

Lemma 7.1 There is a sequence On tending to 0 such that if Pn is an arbitrary polynomial of degree at most n, and

s 1,

(7.2) and

x E Sk,

s 1,

(7.3) then

x Elk,

s On'

(7.4) Recall that Wk(X) = exp(UlIk(x)).

Proof of of Lemma 7.1. We begin the proof by reducing it to the case when Pn has a special form. First of all by considering (Pn(x) + Pn(-x))/2 instead of P« we can assume that Pn is even. Let

Pn(x) = an

II(x

2

II(x

2

-

ai),

i

and

P:(x) = lanl

-Iail).

i

Since IPn(O)1 = and for every x E R the inequality IPn(x)1 holds because of the obvious inequality Ix 2 - ail Ix2 - lail!' without loss of generality we can assume Pn = i.e. that the leading coefficient an is positive, and aj 0 for all i. which also means that Pn has only real zeros. Let o al, ... , a 3 < 2- 2k and a 3 +! , ... 2- 2k , and now consider

and for every Ixl 2- k the inequality IPn(x)1 Again Pn(O) = holds because for such x the function Ix 2 - alia decreases in a on the interval [0,2- 2k ) . Thus, (7.2)-(7.3) hold also with Pn replaced by while the status

40

Approximation with general weight.

Section 7

of the inequality (7.4) remains unchanged, hence without loss of genrality we can assume Pn = i.e, that Pn does not have any zero on the interval (_2- k,2- k ) . Let M be arbitrary large. We distinguish two cases. Case 1. There are at least M zeros of Pn in I». If we set

Un(x) = log

- nUtJk(x),

then (7.2) takes the form (7.5) for every x E 8k = SUpp(Vk)' We can apply the principle of domination (see the proof of Lemma 5.1 in Section 5) to deduce (7.5) for all x E C. Now let us recall Harnack's principle, according to which if U is nonnegative and harmonic on a domain D, and Yl, Y2 ED, then there exists a K independent of U such that U(yd KU(Y2)' We set D = (C \ R) U (_2- k , 2- k ) . Since all the zeros of Pn lie in R \ (_2- k , 2- k ) , we can conclude the harmonicity of Un in D, and so for Yl = 0, Y2 = 00 there is a K such that (7.6) independently of n. Now integrate (7.5) with respect to the equilibrium measure WS k of 8k. Taking into account (1.7)-(1.8) we get from Fubini's theorem with the counting measure lin on the zeros of Pn 1 nlog cap (8k )

where, at the very last step we used (7.1) and the fact that at least M of the zeros of Pn lie on Ik. This implies that (7.7) and so, in view of (7.6), we can conclude

Case 2. There are at most M - 1 zeros of Pn in I k . Let Js, ... , J M be disjoint open subintervals of Ik and t1, ... ,tM one-one point in each. Choose connected open sets Dj, j = 1, ... , M containing the points 0 and tj such that

o, n R S; s, U (_2- k,2- k ) .

Section 7

41

Construction of Examples 4.5 and 4.6

By Harnack's inequality there are numbers K j such that if U is a nonnegative harmonic function on Dj, then

Now in this case there are at most M -1 zeros on Ik' so these zeros have to miss at least one Dj , say Dj " , which means that Un is nonnegative and harmonic on D j " . Hence the preceding inequality can be applied with U = U« and j = in, by which we get from the assumption (7.3) that

Un(O)

l. K Un(tj,,) J"

l. K nlog,B J"

n (._min

J-l, ... ,M

l.) K 10g,B. J

This gives us for large n

Un(O)

M,

which, together with (7.7), proves our lemma because M can be arbitrary large.

I Having this lemma at our disposal, we are proceeding with the construction of

Vk+l'

Let us choose and fix an nk such that ()nk < l/k (see the previous lemma), and let Pk denote the set of polynomials Pn of degree at most nk that satisfy

(7.8) This set of polynomials of a fixed degree is compact in the supremum norm, hence we get from Lemma 7.1 that there are finitely many points tl, ... , tN on Ik such that if for Pn E P» the inequalities (7.9)

are satisfied for every j, then (7.10)

Let H k be a subset of Ik containing the set {tj }f=l such that it is symmetric with respect to the origin, it consists of finitely many intervals, and cap(Hk) is some Ck that will we selected below together with the positive number Pk. If Wk := WHk is the equilibrium measure on Hk' then the potential of the measure p"Wk is nonnegative on [-1, 1] (by the symmetry of Hk), it is of the order O(Pk) on Sk, at each tj it has the value PIc 10g(l/c,,), and at every other point of [-1,1] its value is at most PIc 10g(l/ck). Thus, by appropriately choosing c" and Pk (to have, in particular Pk 10g(l/ck) = 10g,B), we can achieve that the weight wk = w" exp(UPkWk) will satisfy the property that

(7.11)

42

Section 7

Approximation with general weight.

implies

(7.12)

(wi;tk(O)IPnk(O)1

(see (7.8)-(7.10)), furthermore Ilog(wk) - 10g(wk)1 10g,8. The point is that by adding the measure PkWk to Vk we get a measure the potential of which is approximately the same as that of Vk on Sk (this is achieved by chosing Pk sufficiently small), but at the points tj it is larger than the latter one by the amount 10g,8, and otherwise on [-1, 1] these two potentials differ by at most 10g,8. Furthermore, the argument also shows that there is a Pk < 2- k such that for every 0 < Pk < pi there is an appropriate Ck (say Pk log cs = 10g,8) with the property that for the corresponding wi; (7.11) implies (7.12). Now there is an fk > 0 such that if w is any function satisfying

Ilog(w(x)) -log(wi;(x))1

fk

for every x E {O} U Sk U Hk, which is a subset of {O} U Sk+l, then

(7.13) implies

(7.14) Let hk be the density of Wk. The function Vk + Pkhk is almost the desired Vk+l, except that it has the following deficiencies: its integral is 1 + Pk, that is slightly larger than 1, it has infinite singularities at the endpoints of the subintervals of H k, and it is not everywhere positive on Sk+l. But all these are easy to rectify. In fact, it easily follows from the monoton convergence theorem and Dini's theorem about the uniform convergence of an increasing sequence of continuous functions provided the limit is continuous, that if we take hi := min(h k, M), then for large M the potentials of this new function and that of Wk differ as little as we wish. Then we can distribute the mass J(h k - hi) that we saved in replacing hk by hi to Sk+l \ (Sk U Hk) so as to get a continuous continuation of Vk that is smaller than 1/(k + 1) if Ixl 2- k and takes the value 1/(k + 3) at the points ±2- k- 1 . In other words, there exists a continuous function Vk+l with the following properties: it coincides with Vk on Sk, coincides with hi on H k, continuous and positive on Sk+l, zero outside Sk+l (recall that U u, C Sk+d, is smaller than 1/(k + 1) if Ixl 2- k , takes the value 1/(k + 3) at the points ±2- k- 1 , and it has integral 1 + Pk. Now

s,

with some small pi > Pk > 0 clearly satisfies all the properties that we required of VI for / = k + 1. Furthermore, this construction can be done in such a way that for the corresponding Wk+l = exp(U tl k + 1 ) the inequality

(7.15)

Ilog(Wk+l(X)) -log(wi;(x))1

s f;,

x E [-1,1],

Section 7

43

Construction of Examples 4.5 and 4.6

is satisfied with the fk chosen before, furthermore

(7.16)

x E [-1,1].

Until now we have not said anything of the constant 13 == 13k > 1. To ensure the uniform convergence of the sequence {Wk} on [-1,1] let us require that log 13k == min(fk-d8, 2- k), which, together with (7.16) yields the uniform convergence of {wd. We can clearly assume that fk :s: fk-d4 and fk :s: 2- k are also true for every k. Then w(x) == lim Wk(X) k-+oo

exists and continuous on [-1,1] (actually on the whole real line), and it follows from (7.15), (7.16) and the inequality log 131 :s: fl-d8 that

(7.17)

Ilog(w(x)) -log(wk(x))1

+L

:s: Ilog(wk+1(x)) -log(wA;(x))1

Ilog(wl+1(x)) -log(wI(x))1

:s:

I>k

for every x E [-1,1]. Our construction gives that the sequence compact subset of (-1, 1), and if

v(x) == lim

k-+oo

2

Vk

+

L fl-1 < fk

4 1> k

uniformly converges on every

Vk(X),

then v is positive on (-1, O)U(O, 1), continuous on [-1,1]' v(O) == 0, furthermore

w(x) == lim Wk(X) == lim exp(Utlk(x)) == exp(Utl(x)) k-+oo

k-+oo

for at least every x E (-1,1). Thus, by Lemma 5.1 we get that the equilibrium measure associated with w will be v( x )dx, i.e. v is the density of /Lw. Thus, it has left to prove that if a function I is uniformly approximable on [-1,1] by weighted polynomials of the form w n Pn , then I must vanish at the origin. In fact, without loss of generality assume that III :s: 1/2. Let 1/ < 1/2. If approximation is possible, then for large enough k there will be polynomials of degree at most nk such that II - wnkPnkl:s: 1/ on [-1,1]. But then (7.13) is true in view of (7.17), hence (7.14) holds. Thus, we can conclude

1/(0)1

s Iwnk(O)Pn(O)1 + 1/(0) -

wnk(O)Pnk(0)1

s

+ TJ,

and since here TJ > 0 is arbitrary and k can be any large number, it follows that

1(0) == 0 as we have claimed.

I In the previous construction we have considered the approximation problem on [-1,1]. To get an example when w is defined on R all we have to do is to extend w to R \ [-1, 1] so that it be be admissible in the sense of (1.3) and be smaller than exp(U tl) there.

44

7.2

Approximation with general weight.

Section 7

Example 4.6

In example 4.6 we have to construct a weight won [-1,1] such that the support of the corresponding extremal measure fJw is [-1,1]' fJw has continuous density in (-1,1) which vanishes at the origin, and still every continuous function I that is zero at ±1 can be uniformly approximated by weighted polynomials of the form w n Pn . Here, as opposed to Example 4.5, we will construct a v which tends to 0 at the origin very slowly. Let {lj} be a countable system of continuous functions that vanish at ±1 such that the linear combinations of {If} is dense in C o( -1,1). Then it is enough to approximate each Without loss of generality we may assume that each Ii is nonnegative smaller than one at every point of [-1, 1]. Let 1 1 w(x) = - Vf=X22 11" 1- x

IJ.

be the so called arcsine distribution, which is nothing else than the equilibrium distribution of the interval [-1,1]. For each n let W n be a function that is continous and nonnegative on (-1,1), coincides with won (-1, -1/n)U(1/n,1), vanishes at 0, it is at most 2 on [-1/n,1/n], and has integral 1. Choose 1 < a < 2, and let

With some sequence {nk} to be chosen below we set oc

(7.18)

v(x) = 'Y Ll-wn / . 1=1

Then v will be a continuous function on (-1, 1) that is positive everywhere except at the origin, where it vanishes. We will show that by appropriate choice of {nk} the weight (7.19) w(x) := exp(UII(x)) satisfies all the requirements. Since Lemma 5.1 easily implies that the density of fJw is exactly v, all we have to show is that every continuous function that vanishes at ±1 can be uniformly approximated by weighted polynomials w n Pn . We shall determine the sequence {nk} recursively along with two other sequences {Nd and {Md tending to infinity. Suppose that n1, ... ,nk-1, N 1, ... ,Nk-1 and M 1, ... ,Mk-1 have already been chosen, and for m 2 let

Section 7

45

Construction of Examples 4.5 and 4.6

where

fA

=

f: 1-0:

and

'Yk

=(

l=k+2 and

Vk,oo(X) = 'Yk

L

1-0:) -1

,

I¢k+l k- l ) L I-O:wn, + k-O:w + 13k w (

.

1=1

Then, if we set Wk,m = exp(Utl k , ,,. ) , it follows from Lemma 5.1 that PWk,m has density Vk,m (note that Vk,m has integral 1). The number m stands for the next nl, i.e. for nk. We shall choose m only at the very end of our construction. Our aim will be to get estimates that do not depend on the actual choice of m. Let us follow the considerations of Sections 2 and 6. By the method of Section 2 the potential nUk,m of nVk,m can be approximated by potentials corresponding to the sum of n Dirac measures. We have to watch however the size of the remainder terms and have to carefully chose L because we want to get an estimate which is uniform in m. The details are as follows. Let f> 0, and consider the set J, = [-1 + e, 1 - f] from Section 2, and the intervals Ij discussed there for the function Vk,m. The best constants c, C for 1= 0 can be easily seen to satisfy c e which c/n IIj I C [n for every t, n and C 41Tk0:- 1 , which follow from

1 1-0: 2 -4 k 2{ can stand in (8.3) and (8.5) instead of Jz«, furthermore in (8.3) we do not actually need geometric convergence, i.e. (8.3) can be replaced by

°

(8.6)

IRm(x) -

11 = 0(1)

for x E J.."

as m -> 00. It is also clear from the proof that (8.5) can be replaced by the uniform boundedness of R m (z ): (8.7)

IRm(x)1

sC

for all m and x E J* .

°

Thus, it is enough to show that in the present case for arbitrary "I > and c > we can choose an e > 0, such that (8.4), (8.6) and (8.7) hold for some polynomials R m of degree at most m whenever m is sufficiently large. The assumption that [0,1] \ 0 has zero capacity implies that the capacity of J* \ J, tends to zero, hence our claim follows from the next lemma by setting S = J.., and J{ = J* \ J..,/2 if we apply it to the sets L = J* \ J, with { < "1/2, e -> 0. Thus, the verification of the lemma will complete the proof of Theorem 8.1.

°

°

Lemma 8.2 Let Sand J{ be two disjoint compact subsets of [0,1]. Then there is a constant 8 > such that for all compact subsets L of J{ and sufficiently large n there are polynomials Pn of degree at most n such that

(8.8) (8.9)

and (8.10)

IPn(X) I :::; 2, IPn(x) -

x E [0,1],

1)6n '

11:::; (2

xES x E L.

Section 8

Uniform approximation by weighted polynomials with varying weights

Proof. Let

51

m

Tm(z) = I1(z -

Zj)

j=l

be the polynomial of degree m that has all its zeros Zj in L and minimizes the norm IITmllL among all such polynomials (these are the so called restricted Chebyshev polynomials associated with the set L). The classical proofs for the Chebyshev polynomials given e.g. in [51, Theorem III. 26] can be easily modified so that we obtain (8.11) lim IITmllilm = cap(L). m-oo

Since all the zeros of Tm belong to [0,1], we also have x E [0,1].

(8.12)

Let now Sp and K p be the set of points on the plane the distance of which to Sand K, respectively is at most p, and choose p so small that the sets Sp and K p be disjoint. Consider the function Im(z) which is defined to be Ion K p and l/Tm(z) on Sp. This 1m is analytic on Sp U K; and we have the bound

for it, hence by a classical approximation theorem of Bernstein ([52, Theorem 5, Sec. 4.5, p. 75]) there is a T < 1 and there are polynomials Rk of degree at most k such that E Sp/2 U K p/ 2.

Z

We set here k = rm, where r is so large that r" (8.13)

Z

< 1/4C1 holds. Thus,

E Sp/2 U K p/2 '

We also get from the Bernstein-Walsh lemma ([52, p. 77]) that with some constant C 2 1 (8.14)

x E [0,1]

c«

1). (note that here We have already used in (6.13)-(6.15) the consequence of (15, Theorem 3] that if there are two disjoint systems of subintervals of [0,1], then there are and 1 on [0,1] and geometrically polynomials that take values in between converges to zero respectively to 1 on the two systems of intervals. Thus, we can choose a '" and for all sufficiently large I polynomials Ql of degree at most I such that IQ1(X)1 ",', x E [0,1] \ SP/2,

°

and

IQ1(X) -

11 < ",I,

xES.

52

We set here I = sm with an s such that

(8.15)

Section 8

Varying weight.

IQ/(x)1

K

S

< 1/2C3 holds, by which we get x E [0,1] \ Sp/2

m ,

and

(8.16)

IQ/(x) -11
0 are nondecreasing on (0,1) that may depend on [a, b], and the functions (d) (c) + 1/ and there are points 0 < c < d < 1 and an 1/ > 0 such that for all n. Then every continuous function that vanishes outside (0,1) can be uniformly approximated on [0,1] by weighted polynomials deg Pn n.

Section 8

Uniform approximation by weighted polynomials with varying weights

53

Being uniformly in C 1+f means that the derivatives satisfy uniformly a Lipshitz condition -

yl',

:::; Clx _

x E [a,b], y E (0,1)

with constants C and e = ia,b > 0 independent of x and y. Note that our assumptions require only Cl+f smoothness on Qn inside (0,1). Pn --+ f holds uniformly on some larger set We can conclude again that [-(), 1 + ()], () > 0 (provided of course the weights are defined there). W n = exp( -Qn) is a sequence of even weights such that the extremal support SW n is [-1,1] for all n, and on [0,1] the functions satisfy the conditions of the preceding theorem. Then every continuous function that vanishes outside (-1, 1) and also at the point 0 is the uniform limit on [-1,1] of weighted polynomials deg Pn :::; n.

Corollary 8.4 Suppose that {w n },

What happens around 0 (i.e. what is the situation ifthe function to be approximated does not vanish at the origin) is quite complicated (see Theorems 12.2 and 12.3 for more details): if wn(x) for all n is the Freud weight exp( IxIO ) , a> 0, then clearly all the conditions of the Corollary are satisfied, but an(y) f that vanishes outside (-1, 1) but not at the origin is approximable by weighted polynomials if and only if a 1 (see [30], and also the discussion in Section 12). We begin the proof of Theorem 8.3 by a lemma.

=

=

Lemma 8.5 Let w(x) exp(-Q(x)) be such that Sw [0,1] and that tQ'(t) nondecreases on (0,1). Then the density of the equilibrium measure dj.tw(t) = v(t)dt is given by (

8.17

)

1 t ( v ) = 1r2

t' sQ'(s) -

tQ'(t) 1 d D s- t Js(l- s) s + Jt(l- t)'

Vt-t- 10

1 11 [6

where

D= - - -2 1r 1r and here D

1

0

--Q'(s)ds, 1- s

O.

Proof. Let f(x) = Q(x 2 )f 2, x E [-1,1]. It was shown in [28, Lemma 5.1] that the integral equation

r

-1

log _I_1_,g(t)dt = - f(x) X -

t

+C

where C is some constant has a solution g(t) of the form

_ 2

g(t) - 1r2

1- t

t' 10

sl'(s) - tl'(t) (1 _ 82)1/2(s2 _ t2) ds

D1

+ v'f=t2'

54

Section 8

Varying weights

where

D1 =

.!. _ 1r

J.. 1 1

1r 2

sf'(s) ,

-1

furthermore g is even and has total integral lover [-1,1]. If we set h(t) = g(0)/0, t E [0,1] then h will have total integral lover [0,1], and by the symmetry of g its potential satisfies

1

1 log -,- , h(u)du o x- u 1

2

1 1

o

1

log , 2,g(t x- t

-2f(vX)

2)dt

1 1

=2

-1

log

.

IJXX1-

t

Ig(t)dt

+ C = -Q(x) + C

for every x E [0,1]. On applying Lemma 5.1 we can conclude that dj.tw(t) = h(t)dt i.e. h(t) = v(t) provided we can show that h is nonnegative (to be more precise the equality dj.tw (t) = h(t)dt automatically follows from the properties above, but we shall need to prove the nonnegativity of D anyway, and this easily implies the nonnegativity of h). The nonnegativity of h will follow from the last statement of the lemma that we are going to show in a short while. In fact, we can see by integration by parts that the two constants D and D 1 are the same, so the relation D is the same as D 1 0. Furthermore, the first term in the expression of g is nonnegative in view of the fact that sQ'(s) nondecreasing. Hence, the nonnegativity of D implies that of h, and this, as we have seen before, implies v(t) = h(t). Now if we carry out the substitution f(s) = Q(s2)/2 and u = s2 in the formula for g, we obtain the form of v stated in the lemma. Thus, it has left to show that D 0, i.e.

°

(8.18)

-

1r

- l - Q'(s)ds

0

-

S

1.

First we show that for every t E (0, 1)

J..l

(8.19)

21r

0

1

1 log 1 - t S -

t

I

1

sl/2(1 - s)3/2

ds = 1.

We shall do this by examining the integral If

«t

= 21r

0

1 log 1 - t S -

I

where * indicates that we skip a small (t - e, t during integration. If

( u t)-l Xf,t(U) = { 0 then

1

ds = 1,

+ f)

(0,1) neighborhood of t

t sl/2(1 - s)3/2

if It - u] e otherwise,

Section 8

Uniform approximation by weighted polynomials with varying weights

55

for all s which belongs to the range of integration in If, hence integration by parts yields

11 [6 1

If = -

11'

As

f. ->

--Xf,t(s)ds. 1- s

0

0 this tends to the principal value integral

..!:.PV 11'

r

io

s 1 ds. s-tJs(l-s)

Unsing now that s/(s - t) = 1 + t/(s - t) and that

(8.20)

PV11 _1_

1 ds = 0, o s-tJs(l-s)

for all t E (0,1) (see [39, p. 251, (88.9)]) we obtain (8.19). Since in (8.19) the integrand is negative to the left of 2t - 1 and positive on (2t - 1,1), we obtain for all a < 1 and t E (0,1) the inequality

.2.r 211' io

log

/1s -- tt Isl/2(1 1- s)3/2ds

r _l_i g'(u)duds Jo s - t t

r

Jo 0:

l

I

g/(u) log _t_ du + t -

d

U

1 1

t

g/(u) log 1- t du

g'(u)du = o:(g(d) - g(c))

U -

t

0:"1

with

I The Corollary immediately follows from Theorems 8.1 and 8.3 (applied to the interval [-1,1] rather than to [0,1] and to the set 0= (-1,0) U (0,1)) if we use the substitution applied in the proof of Lemma 8.5.

Section 9

9

Modification of the method

57

Modification of the method

In this section, we modify the method that we used above. This modification allows us to get better aproximation around the endpoints of the interval. These are needed if we want to handle infinite singularities inside the support of the generating measure, or if the approximation has a second constraint that frequenly appears in applications (see Theorem 10.1 and 10.2 below). The modification is roughly as follows: In the first two parts of this work we approximated a given potential by first translating the generating measure and then appropriately discretizing it. The idea here is similar, but instead of translation, that can be viewed as projection onto a segment, we shall project it onto a curve, which is closer to the support exactly where the generating measure is larger. We start with a technical lemma. Lemma 9.1 Suppose that {Un} is a sequence of nonnegative functions on (0,1) satisfying the following conditions: {un} is uniformly equiconiinuous on every compact subinterval of (0, I),

1 1

Un = 1,

and for some constants A> 0, 13

(9.1)

Un(t)

(9.2)

un(t)

5

> ­1

and 130

A(t(l ­ t))13, (t(l ­ t))13o,

tE(O,l),

t E (0,1).

Then there is an L o > 1 such that for every L > Lo there are polynomials Qn of degree at most n such that for large n, say n nL

(9.3)

o

5

log IQn(x)1 + nUU,,(x)

< {BL

2

10g n

BL 2log l/(x(l ­ x))

if x E [0, n- 1] U [1 ­ n- 1 , 1] if n -1 5 x 5 1 - n -1,

and with some continuous functions gn that are uniformly bounded and uniformly equicontinuous on compact subsets of (0,1)

Here B is an absolute constant depending only on A,13 and 130' The same conclusion holds if we assume the inequality in (9.2) only on the interval [n- T , 1 - n"].

The lemma is true in more general situations (allowing several intervals or zero or infinite singularities in the weight), we shall comment on that after the proof.

58

Varying weights

Section 9

Proof. We shall only concentrate on the behavior on the interval [0,1/2]' the other half being symmetric. We shall indicate at the very end of the proof the necessary modifications we have to make to cover the whole interval [0,1]. Without loss of generality we can assume that (3 < 0 and (30 > O. Fix a constant L. We select a continuously differentiable function V n that has the same size as Un on [L-9,1 - L -9], as follows: for x E [L-10,1 - L -10] we set x dL + un(t) dt, (9.5) vn(x) = 2d1

l

X-dL

L

where the constant dt. satisfies

for all n (such a dL exists because the functions Un are uniformlyequcontinuous and uniformly bounded below by a positive constant on compact subsets of

(0,1)).

Let 0 < TO < 1/((30 + 1) if (9.2) is true for all t E (0,1), and 0 < TO < min{ T, 1/((30 + I)} if we assume (9.2) only for t E [n- T , n T ] (in the former case we may set T = 00). For an n let us choose consecutive intervals Ij, j = 0,1, ... , n - 1 starting at 0 such that on Ij the function Un has integral equal to l/n:

j

Un

n

I;

As in Section 2 let

be the weight point of

=n

f

JI;

Un

tun(t)dt.

Let j = 0,1, ... , N be those indices for which Ij (9.7)

IIj I

mj

on Ij:

= { II; un(t)/vn(t) dt

[0, L -9]. Set now

if j = 0, 1, ... , N if j > N.

We claim that the polynomials n-1

Qn(x) =

+iLmj) j=O

satisfy all the requirements of the lemma. Before we embark on the proof, we make a few preliminary observations. In doing so let us agree that in what follows c, C denote positive absolute constants depending exclusively on A, (3 and 130. However, we allow C and c to change from line to line. As before, the symbol R'" S will indicate that c :0:; R/S :0:; C with some such c, C.

Section 9

59

Modification of the method

It follows from assumption (9.1) that (9.8)

ej

1)

j + c ( -n

1/(1+13)

and for all j

IIjl

,

c(j + 1)-P/(1+P)n- 1/(1+P),

rn·,...., II-I' 11

(9.9)

0$ j < n.

(9.2) gives (9.10)

m1\X IIj I 1

s C max{ n- r , n- / (Po+1) } $ cw:» 1

regadless if (9.2) holds for all t or only for t E [n- r , 1 - n- r ]. The choice of 'To implies via (9.10) that if x E [n- ro , 1/2] and I is an interval oflength x/8 that intersects (x/2, 2x), then

i1

{ Un

> .!.. n

This means that there is at least two-two Ij lying stricly in the intervals (x/2, x) and (x, 2x). For the length of every such an Ij we obtain from (9.1) and (9.2)

t,

(9.11)

(x/2, 2x).

Finally, we mention the formula (9.12) The lower estimate

First we prove that

log IQn(x)1 + UU,,(x)

O.

In view of (9.12) it is enough to show that n { iIj

IOglX -ei +iLrnj /un{t)dt X -

t

0

for all x and j. Let x E Ijo' If j = jo, then the quantity under the logarithm is at least as large as

I

L x

I

t

L

1,

for sufficiently large L, where we used (9.9). If, however, j f:: jo, then the quantity under the logarithm is at least as large as

I

x -

ej I,

x -t

60

Section 9

Varying weights

and by the concavity of the logarithm function log Ix - tl for t E Ii we get

n 110g Ix - tlun(t) dt $ log Ix - ei I· J

These inequalities prove the lower estimate. The upper estimate We distinguish two cases according to the location of x. Case I. x E [0, n- TO]. Let x E Iio' We divide the sum in (9.12) into three ranges: io+l

L

+

L

+

L =L

1 + L2 + L 3 '

iio+l We estimate these three sums separately. In L2 we have (at most) three terms. The second one is at most as large as

n { IOglx-eio+iLmiOlun(t)dt$n { 11jO x- t 11jO

1

(I

2 Iog((1 + C 2 L 2)IIio I) + n 1

$

x- t

1JO

log Ix _ tl un(t) dt.

We break the last integral into two parts, for integration over Ix - tl $ n-(1+P) and the rest. In view of (9.1) we can get the following upper bound for this integral:

1

1

n -(1+/J)

n

(log

$ Cntl+131

t) At 13 dt +

t=n-(l+/J)

log n(1+I3)

log n(1+I3)

+ (1 + {3) logn $

Clog n.

This shows that the second term is at most Clog Ln, and similar estimates hold for the first and third terms. Thus, altogether we have

L s ClogLn s CLlogn. 2

Now let us consider

L3'

For j > jo we set

Ai = IIio+ll + ...+ IIil· Then for t E Ii we get from (9.9) log x -

I

e-J + iLm'l J X -

t

< log JAi + iCLIIil1. Ai-l

1 ( 1 + C 2 L 2 11'12) = log -A· .J_ + -log

AJ -

1

2

Ai

•

Section 9

61

Modification of the method

Thus,

"M1 AJ 1 ,_

=" L-iS1 +" L-iS2.

Now the analogue of (9.13) is "c:

n 1

31

A = log -A-M

1-1

1 4 s log -A-s log -, X Ml-1

62

Section 9

Varying weights

while that of (9.14) is

L s CL L 2

32

IIjl

j>Ml

s CL 2

lId

1 1

AM1-l

These prove

t

dt

s CL 2log 2.. x

1

2Iog-, '" - CL LJ 3 < x

and the estimate of L1 is analogous. Now let us consider L2. We write with 6 = 2(P - Po)

where the range of the summations in L21 is i = io ± 1, in L22 it is 1 < Ii - io I x- 6 , while in L23 it is x- 6 Ii - io I, and recall that in every case we have Ij (x/2,2x). In L22 we can utilize once more the technique of (9.13) and (9.14). In fact, if M2 is the largest integer that is smaller than x- 6 , then the part of L22 which corresponds to the indices io + 1 < j i« + M 2 can be estimated with the method of (9.13) and (9.14) as

Here we can use (9.11) according to which

and A j o+1

1 _ A(4x) Pin,

These imply via the preceding formula

",' < CL 2 Iog ! . LJ23 -

x

The other part of L23 (corresponding to the indices io be similarly handled and we arrive at

M 2

5 j < jo - 1) can

1

'" CL 2 Iog -X . LJ 23 < In L21 we have three terms. As before, we get for the middle one

63

Modification of the method

Section 9

1.

10

Since the integral of Un on the interval I; is lin, there must be a point t1 E I; such that un(td 1/nII;I. But for any tEl;, I; (xI2,2x), the ratio un(t)/un(td is less than CA 2xfJ-fJo (see (9.1)-(9.2)), hence for any tEl; we have nun(t) CA 2xfJ-fJo111;1. Thus, if we break the last integral into two II;lx 6 and the rest, then we can deduce the parts, for integration over Ix - tl following upper bound for this integral: 2 XfJ-fJo 6 log -1 + CA

II; I

x 1

1

II· 0 I log _3_ _ dt

Ix - tl


130 - 13), and similar estimates hold for the first and third terms. Thus, altogether we have 1

'" < CL 210g-. LJ21 X Thus it has left to deal with the sum use the inequality: if 77 -1/4 or if I(JI (9.16)

Ilog11 + 77 + i(Jl- 771 =

When tEl; then It 10gIX-

e; I

2:23'

For the j's in

10g(1 + 277 + II; I

2:23

we shall make

1771, then +

(J2) -

771

2(772 +

(J2).

Lnu , and so by the preceding inequality

e; +iLm; I=log!l+ t-e; +iLm; Is t-e; +2It-e;12+;2mi, x-t

x-t

x-t

(x-t)

The first term on the right is

Thus, if we multiply the last but one inequality with nUn(t) and integrate on Ij, then by taking into account the fact that the integral of the term

vanishes because of the choice of ej, we finally get that

64

Section 9

Varying weights

Taking into account that by (9.11) here

while dist(x, Ij)

(j -

Aj-1

io -

1) (4x)-J} In,

we can continue the last estimate as

by the choice of fJ = 2(/30 - /3). The inequalities that we have proved so far verify the upper bound in (9.3) for x n- To . With this the proof of the upper estimate in (9.3) is complete.

The asymptotic estimate Let x L- 6 . We are going to verify (9.4). Let i = 0, ... , N 1 - 1 be those indices for which

and let I be the union of the rest of the Ij's:

t..

I = Let

be the restriction of Un to I, and similarly, let

n-1

=

II (x -

+ iLmj),

j=N 1

i.e. in we only keep those terms from Qn that correspond to indices for which Ij is lying in I. First we remark that

o < =

(log IQn(x)1 + nUu,,(x)) - (log

1: iI;f

+

1

j=o

+iLmj IUn(t)dt.

x- t

Here the terms under the integral sign are bounded by Iog 11 +

2)IIjI2

x-t

2)IIjI2

C(I+L C(1+L _-+ . < --+ . , x-t dlst(X, Ij)2 dlst(X, Ij)2

65

Modification of the method

Section 9

where we used (9.16). The integral against 'Un(t) dt of the first term on the right vanishes again because of the choice of e;, hence,

(9.17)

0::::; (log IQn(x)1 + nUUn(x» - (log

< CL 2 -

11;12 < CL 2 LJ dist(x 1·)2 ;=0 ' 3

LJ ;=0

+

1;1) 2 < CL 1 2/x -

2

(L-

S

L-6

) 2

0 intervals I j lying in E [L- 7,L-6]. We can write this difference in the form

n-1 En

. N

1=

1

1 I· J

I

log x-

t +'L (t)-l X

1

+

-1

.n

m1

Iun(t)dt


polynomials Rn-[n/>...] of degree at most n - [n/ An] such that for sufficiently large n the following estimates hold with some absolute constant CL independent of q and n:

°

= (x(1- x»)'Y

lim Rn-[n/>. ](x)eLg..

u(x)

n .....o o . .

uniformly on [L-6

+ q, 1- £-6 1

q],

Rn_[n/>. ..](x)eLg..

CL

on the intervals [£-6, £-6 + q] and [1-L- 6 1

-

q, 1- £-6], and

s Rn-[n/>...](x) s 2

on [0, £-6] and on [1- £-6,1]. In fact, the functions rn(x)

== if x E [£-6 + q, 1- £-6 - q] ifxE[O,L-6]U[1-L-6,1] on [£-6, L -6 + q] and on [1_£-6_ q,1_L-6]

e-Lg"(")w;6"(x)(x(1- x»"f /u(x)

1 { linear

°

form a compact subset of e[O, 1] (they are uniformly bounded and uniformly equicontinuous in n), hence for every e > there is an m such that for every n there are polynomials R:n,n of degree at most m such that em,n :=

IIrn

-

e.

R:n,nIlC[O,l]

This means that if we choose Rn-[n/>...] as Rn-[n/>...](x) :=

..],n(x)

+ {n-[n/>' ..],n + e-Ct/L,

with some appropriate but fixed C 1 (that depends only on the constants in the preceding inequalities), then (by n - [n/An ] nP/2) this choice will satisfy the above inequalities. Thus, althogether we get for the polynomials

of degree at most n for every large n, say n 1

for £-6 + q

x

s

1- L-6 - q,

nL

the estimates

x»-"fu(x)

s eCL-

1

74

Varying weights

for L- 6

x

L- 6 + q and L-6 1

for n- T

x

Section 10

L- 6 + q,

1- x

s

s (X(1

L-6 and n- T 1

x))

L- 6, and

1- x

s

s

for 0 x n- or 0 1 - x n- , where the constant C is independent of q and Land n nL,q' Since these inequalities actually give estimates on h« (see (10.3)), and they 1, all we have to verify that the geometric means in (10.5) are show that hn as small as we like. The preceding estimates give for this mean and for large n the inequality T

o
,,,[n/>',,lIQ[n/>',,l(x)!

s1

for all x E [0,1],

=

1- x

for 0 x n- T or 0 and the functions

= Lgn(x)

nr", where the 0 is uniform in Land n

»i:

+ N log((x + 1/n)(1- z + lin))

are uniformly bounded and uniformly equicontinuous on [L-6, 1- L- 6]. With the argument applied in the preceding proof we can find polynomials Rn-1-[n/>',,1- 2N of degree at most n -1- [nl An ] - 2N such that for sufficiently large n the relations

uniformly on [L- 6 + q, 1- L- 6

-

q],

1

on [L- 6,L-6

+ q] and

[1- L- 6 - q, 1- L- 6], and 1

Rn-[n/>'"l(x)

1

2

on [0, L -6] and on [1 - L -6,1], hold with some constant CL independent of n and q. We set Hn-1(x) = Q[n/>',,1(x)Rn_1_[n/>',,1_2N(X)e- C t/ L , which has degree at most n. The estimates

76

Section

Varying weights

for L- 6

+q

x

1- L- 6

-

q,

1

x)l(x(l- x))-'Yu(x)

for L- 6

L- 6

x

+ q and

10

L- 6

1- x

1

L- 6

(x(l- x))2DL

2

+ q,

x))-'Yu(x)

(x(1 _ x))2DL

2

for 0 x L -6 and 0 1 - x L -6 hold with some constant CL independent of q and n nL, and some absolute constant C. These tell us first of all that h n 1 (see (10.13», and then that

o> -

1 1

0

log h n(x) dx > -C ( L 2 y'x(l-x) -

1

L

0

-6

log l/x dx y'x(l-x)

+ q log C + L - 1) L

.

Choosing first L large, then q small we can make the right hand side as small as we wish. The proof is complete.

•

In order to be able to apply Theorems 10.1 and 10.2 we need convenient criteria in terms of the weight w itself (note that these theorems refer to the density function). In the rest of this section we discuss what smoothness conditions on w ensure that the assumptions of Theorems 10.1 and 10.2 are satisfied. We shall do this with conditions similar to those in Theorem 8.3.

Theorem 10.3 Suppose that {w n}, W n = exp( -Qn) is a sequence of weights such that the extremal support SW n is [0,1] for all n, the functions are uniformly of class C" on [0, 1] for some e > O. Suppose further that the functions are non decreasing on [0,1] and there are points 0 < c < d < I, and an 1] > 0 such that for all n. Then the conditions of Theorems 10.1 and 10.2 are satisfied and their conclusions hold. For example, the conditions of this theorem are true if all a single C" function, say if Qn(t) = t Oi for an Q' > O.

coincide with

Proof. We use the representation (8.17), and set gn(t) = C" property of the gn's easily imply that the integrals

The uniform

In(t) =

1 1

o

gn(S) - gn(t) 1 ds s-t y's(l-s)

are uniformly bounded by a constant multiply of (x(1 - x))-1/2+f/2, hence condition (10.1) is true. We have already used in the proof of Theorem 8.3 that the uniform C" property of the gn 's implies that ofthe integrals In (t) on compact subset of (0,1) (see e.g. the Plemelj-Privalov theorem in [39, p. 46]), and this is more than

77

Approximation in geometric means

Section 10

the uniform equicontinuity of the densities Un of the corresponding extremal measures J.lw' Therefore, it has left to verify condition (10.2). But we have seen in the proof of Theorem 8.3 that the integrals In (t) are uniformly bounded from below by a positive constant, and this implies (10.2) via the formula (8.17).

I

We shall also use the following corollary (d. Theorems 10.1 and 10.2). Corollary 10.4 Suppose that {w n } , W n = exp(-Qn) weights such that the extremal support SW n is [-1,1] for functions satisfy the conditions of the preceding theorem. is any positive continuous function on [-1,1]' then there degree at most n such that for

is a sequence of even all n, and on [0,1] the If 1 'Y 0 and u( x) are polynomials H n of

we have

hn(x)

(10.17)

1,

x E (-1,1),

lim hn(x) = 1

n ...... oo

uniformly on compact subsets of (-1,1), and

· 11m

n_oo

In a similar fashion, if 'Y and

11 0

loghn(x) d - 0. r:;---::rr x v1- x-

1/2 then the conclusion also holds for some H n -

hn(x) =

1

x 2)--r u(x)

with (10.17) replaced by

xE(-l,l). Exactly as in the case of Theorems 10.1 and 10.2 the degree of Hn can be somewhat smaller than n, namely we can have deg(Hn) = n - in where in ...... 00. Proof. This corollary is an immediate consequence of Theorems 10.1 and 10.2 and the discussion made at the end of Section 9. In fact, by using the transformation x ...... x 2 applied in the proof of Lemma 8.5 we can see that the assumptions of Lemma 9.1 hold true on (-1, 0) and on (0, 1), and at the end of Section 9 we mentioned how to use these facts to conclude the statement of Lemma 9.1 for the union of these intervals. Now the proof of Theorems 10.1 and 10.2 were based on Lemma 9.1, hence the proof can be copied to yield the corollary.

I

Part IV

Applications In this chapter we shall briefly discuss some applications of our results. They are here for illustration. Some of these have been achieved in less generality by different authors using different techniques, we shall give proper reference at those places.

11

Fast decreasing polynomials

In this section, we shall discuss an application of the method developed in the preceding chapters. We call polynomials Pn , deg-Pn n, fast decreasing on [-1,1]' ifthey attain the value 1 at x = and decrease fast away from the origin:

°

(11.1) We shall discuss the problem with what and n this is possible. The significance of such fast decreasing polynomials lies in the fact that they approximate the "Dirac delta function" as best as possible among polynomials of a given degree, hence for example these are the best polynomial kernels for convolution operators to reproduce the identity. By integration we can get from the above polynomials good polynomial approximants Sn of the signum function in the sense (11.2) [signz - Sn(x)1 x' E [-1,1], which in turn can be used to construct well localized "partitions of unity" (c.f. the construction on p. 156 in [14]) consisting of polynomials of a given degree n. The problem can be formulated in two different ways: one can ask what possible decrease (i.e. what 0 if and only if

du < 00.

f1

10

u

In [49] a potential theoretical method was developed for obtaining sharp results for the largest possible c in (11.3). It was shown there that if r.p is even, increasing on [0,1], and r.p(y'X) is concave on [0,1], then there are polynomials satisfying (11.3) only if

r

1r10

r.p(t)

dt < 1 -

holds, and if we have here strict inequality then such polynomials do exist. This can be applied to any r.p(t) = cltl", 0: 2 and we obtain that there are polynomials Pn with (11.4)

Pn(O) = 1,

only if 0: > 1 and c J7rr Ir ("2 1 ) , and conversely, if the strict inequality holds, then the existence of polynomials with property (11.4) was proven in [49]. The existence of the polynomials in question when the equality holds has remained open (it was resolved in [31] for 0: = 2). Now the discretization method of Sections 2 and 3 enables us to settle this problem.

Theorem 11.1 Let if and only if 0:

0:

> 1 and

2. Then there are polynomials P n with property (11.4)

r("2 1 )

'

Let us mention that the latter result is no longer true for a > 2. It is an open problem to determine the largest constant c that allows (11.4) when a > 2.

Proof of Theorem 11.1. The necessity of the condition was proved in 2 and c [49, Theorem 3.3]' so it is enough to prove that if 1 < 0:

J7rr

Ir ("2 1 ) , then there

are polynomials with the property (11.4). Let us consider the energy problem on E = [-1,1] with weight w(x) exp(clxl") (note the positive sign in the exponent which makes this weight essentially different from the Freud weights). First we show that the corresponding extremal measure is given by the density function

(11.5)

81

Fast decreasing polynomials

Section 11

where the constant dOt is

dOt =

t'

io

1 - u 2 - Ot

(1 _ u2)3/2 duo

To get this form let us consider the function

I(x) =

1 1 - x2

0:

1r

-

(0: -

0:

1)-

1

Ot- 1 u 2du 1"1 -/u 2 - x 1

1r

built up from the Chebishev and Ullman distributions (see (3.4)), and recall that the Ullman distribution corresponds to the energy problem with respect to the weight exp(-,OtlxI Ot) (note the negative sign, which is not the case with the energy problem we are discussing now). This function has total integral lover [-1,1], and by (3.6) its logarithm for x E [-1,1] is of the form const + (0: l},OtlxIOt, where

_ f(j)f(!)

t

2r ( Ot 1)

lOt But using that f(t

+ 1) = tf(t)

(0: -l},Ot =

(0:

and r(lj2) = .,fi we easily obtain that

-1)

r (.2:) r (1)

=

.,fif (.2:) (¥) =

f

c,

and so the potential is of the form const + clxl Ot. Thus, if we can show that the function 1 is nonnegative, then we can invoke Lemma 5.1 to conclude that 1 is nothing else than the density of the equilibrium measure in question (actually, the same conclusion can be derived from the principle of domination without referring to the nonnegativity of I, but we shall need the following consideration anyway). Clearly, it is enough to consider positive values of x. If we write the integral in 1 in the form 1/" uOt-1 x Ot- 1 du

f

N-=1

1

and integrate by parts then it follows that ,

x

0:

1 satisfies the differential equation ::¥-1

1 (x) = -; (1 _ x 2)3/2 + -x-/(x) with initial condition 1(0) = O. We can solve this linear equation and get with some constant d that

I(x) = x

Ot- 1;

(1" (1

The value of d follows from the condition that 1 has integral lover [-1, 1]. This means that we must have d 0: u 2 - Ot 1 Ot- 1-; -; + x (1 _ U 2)3/ 2 dudx = 2'

t

io

r

io

82

Section 11

Applications

which easily yields the value da for d. Since da is nonnegative, the nonnegativity of f follows from the preceding expression for f( x), and the same expression verifies (11.5). When Q = 2 then d a = 0 and (11.5) takes the form

v(x)

x2

2

= ;: (1- x2)l/2'

while for 1 < Q < 2 the constant da > 0 and in this case the density v has order r - Ixl a - 1 as x approaches O. Thus, if 8 Q - 1 if 1 < Q < 2 and 8 2 if Q = 2 then we can conclude that the density v of the extremal measure satisfies v(t) '" Itl 6 as t --+ 0 and v(t) '" (1 - t 2)- 1/ 2 as t --+ ±1, and otherwise v is continuous and positive. This is all we Heed of v. Let now jl(t) = v(t)dt be the extremal measure. By Theorem A from the introduction we have UIJ(x) = clxl a Fw for every x E [-1,1]. If we can construct polynomials

=

=

+

n-l

Rn(x) =

II (x - ej)

j=O

such that (11.6)

-log IRn(x)l- nUIJ(x)

C

- log IRn(O)I- nUIJ(O)

C,

for all x E [-1,1], and (11.7)

then Pn(x) = Rn(x)IRn(O) will satisfy (11.4). This is where the discretization technique of Section 2 enters the picture. Let n be an even number (when n is odd, use n - 1 in place of n below). Let us divide [-1,1] by the points -1 = to < tl < ... < t« = 1 into n intervals Ij , j 0,1, ... , n - 1 with jl(Ij) lin, and let be the weight point of the restriction of jl to I j . Set

=

ej

=

n-l

Rn(t) =

II (t - ej)·

j=O

We claim that these satisfy (11.6) and (11.7). We write (11.8) -log IRn(x)l- nUIJ(x) =

n j=O

1 Ij

I

log x x

t.1 v(t)dt =:

e 1

.

I:

Lj(x).

j=O

The proof of (11.7) is very simple: since en/2-1 < t n / 2 = 0 < en/2, and the function log 10 -tl is concave on every I j , we get that every term Lj(x) in (11.8) is at most O. This proves (11.7). It is left to prove (11.6). Let x E Ijo' The individual terms in (11.8) are clearly bounded from below when j = jo, io ± 1. For other j's the integrands

83

Fast decreasing polynomials

Section 11

are bounded in absolute value by an absolute constant independent of n, x and :j: io, io ± 1, hence the integrals themselves are also uniformly bounded, for the integral of v on each Ij equals l/n. As we have done in Section 2, we write for x E Ijo' and i :j: io, jo ± 1 the integrand in Lj(x) as

i

log 1 + _J__ =

I

I

X -

_J_ _

x-

+0

(I

(-t

_J__

x-

2)

1

,

which holds because x-

t > _q > -1

-

with an absolute constant 0

-

,

t E t..

< q < 1. Thus, we have

(11.9)

1, ;j

because the integrals

v(t)dt

vanish by the choice of the points We have to distinguish two cases according as x is closer to one of the endpoints or it is closer to O.

Case 1. x is close to an endpoint. [-1, -1/2]. We have to estimate

Let us suppose for example that x E jo-2

8 1 (x) :=

L

ILj(x)1

j=O

and

n-1

8 2 (x) :=

L

ILj(x)l·

jo+2

We shall only do the first one, the second one being similar (in view of (11.9) the part of 8 2 corresponding to the indices for which > -1/4 is less than

CLlIjl2 s CLIIjl

(11.10)

j

For

i

io -

2

j

s C.)

84

Section 11

Applications

hence

Here

K 0

=

1/(6+1) ei* '" (i) , n

where 6 is the number chosen above, i.e. 6 = Q =2. If x E Iio' io > 0 then we have to estimate

Q -

1 if 1
0 there is a continuous function X (that may also depend on f) such that X(O) = 1, and for all sufficiently large n there are polynomials Pn of degree at most n such that IW(x)Pn(x) e for all x E R. In fact, suppose this is true and we want to x(x/an)1 approximate an f which vanishes outside (-1,1). Then, in view of Theorem 12.1, for a given f > 0 we can approximate f(x) - f(O)x(x) by a W(anX)R,,(x) uniformly on R with error smaller than f. The sum Pn(anx)+R,,(x) multiplied by W(anx) will then be closer to f than 2f. We shall need a lemma, which is a variant of a result of D. S. Lubinsky [24]. Lemma 12.4 Under the conditions of Theorem 12.3 there is an even entire function H with nonnegative McLaurent coefficients such that W(x )H(x) - 4 1 as x - 4 00. Proof of Lemma 12.4. The lemma can be proven with the method of [24, Theorems 5,6]. We shall only indicate how the construction goes and what changes are necessary in those proof. As before, let W(x) = exp(-Q(x)), and for x > 0 let the number qx be the solution of the equation qxQ'(qx) = 2x. Then

(12.2) With the function T( x) from the theorem we set

This H will satisfy the claim in the lemma. The proof is an adaptation of that of [24, Theorem 5], we are not going into the details.

I

Now let us return to the proof of Theorem 12.3. Let Rm be the m-th partial sum of the McLaurent expansion of the entire function from the preceding

90

Section 12

Applications

lemma. Since the coefficients of H are nonnegative, we have for any x the Rm(x) H(x), furthermore, Rm(x) H(x) uniformly on inequality 0 compact subsets of R. As for the Mhaskar-Rahmanov-Saffnumbers an, we can conclude from the assumption (12.1) that there is a positive constant C such that an a2n Can holds for all large n. This and (12.2) easily imply (see also the computation for qo; in [24]) that given any f > 0 there are numbers r and t such that IH(x) - Rm(x)1 for Ixl ram H(x) e

s

and

s

IRm(x)1 < f H(x)

-

for

Ixl

tam'

Now W(x)H(x) -1 tends to zero at infinity, hence by the solution to Bernstein's problem to every e > 0 there exists a polynomial 8 such that for all x ER IW(x)(W-1(x) - H(x) - 8(x))1 f. This gives that

11 -

W(x)(H(x) - 8(x»1

e,

and so in view of of the preceding inequalities we obtain that (12.3) (12.4)

and (12.5)

11- W(x)(Rm(x) -

8(x))l

W(x)IRm(x) - 8(x)1

(1 + M)f

for

Ixl

ram,

Mf

W(x)IRm(x) - 8(x)1

M(M + 2)

otherwise, where M is an upper bound for W H on R. The assumptions on Q imply (see also the computation for qo; in [24]) that there is also an L such that 2ta[n/L] ran is also true. Now if we set

then we ge from (12.3)-(12.5) that

IW(x)Pn(x) - x(lxl/an)1

(1 + M)f +

M(M +2)

1

'

where the function X(x) is defined on [0,00) as follows: X(L- k ) = 1- 11k for k = 1,2, ... ,1-1, X(O) = 1, X(x) = 0 for x 1 and X is linear otherwise. Hence the polynomials Pn and the function X satisfy the requirement from the beginning of the present proof if we choose 1 > M(M + 2)1f.

I

Section 13

13

91

Extremal problems with varying weights

Extremal problems with varying weights

Let {w n } be a sequence of weights on the interval [-1, 1], u a fixed nonnegative function and for a 1 p < 00 consider the extremum problem (13.1) where again TIm denotes the set of polynomials {x m +...} with leading coefficient 1. We have already discussed this in Section 3.3 in the particular case when each W n is the same. From Theorems 10.1 and 10.2 and Bernstein's formula (3.38) we can easily get the following strong asymptotics for Em,p. Theorem 13.1 Let 1 p < 00, {w n} a sequence of weight functions on the interval [-1,1] such that the corresponding extremal measures J.tw,. have support [-1,1], they are absolutely continuous there and if we write dJ.tw,.(t) = vn(t) dt, then the functions V n satisfy the conditions

(13.2)

vn(t)

(13.3)

vn(t)

A(1- t 2 ),B ,

t E (0,1),

(1- t 2 ),Bo,

t E (0,1)

for some constants A, 13 > -1 and 130. Let furthermore u be a positive continuous function. Then for every k = 0, ±1, ...

(13.4)

= (1 + o(1))up2-n-k+l-l/PG[Wn]nG[u]

as n - 00. Actually, the relation (13.4) is uniform in k -K for every fixed K. The result is also true for even weights {w n} satisfying on [0, 1] the conditions of Theorem 10.3. Recall that

G[V] = exp

(l-jl

logV(x)

dX)

1r

are the geometric means from (3.13) and

up = ( f(1/2)f((p + 1)j2)/f(pj2 + 1))

l/P

.

If we integrate the equality in Theorem A,(f) with respect to the equilibrium measure of the interval [-1,1] and use Funibi's theorem as well as (1.7)-(1.8), then we get that G[W n] = exp(1og2 - Fw ,. ) ,

hence (13.4) can be written in the form

= (1 + o(1))up2-k+l-l/PG[u]e-nF..,.. The result holds for somewhat more general u's, but the exact conditions on u are not clear. Below we shall prove a more general result for p = 2. This case corresponds to orthogonal polynomials and appears in several situations.

92

Applications

Section 13

Proof. The proof is very similar to the argument of Section 3. First let us assume that k ;::: O. The assumptions imply that Theorems 10.1 and 10.2 can be applied on the interval (-1,1) rather than on (0,1). We set "I = (1-I/p)/2. Then by Theorem 10.1 there is a sequence {H n } of polynomials of corresponding degree n = 1,2, ... such that if

then hn(x) ;::: 1 and Thus, by Bernstein's formula (3.38)

/ (lTp2-n-k+l-l/PG[Wn]nG[u]) ;:::

En+k,p( ipl/2-1/2p /IH nI)/ (lTpTn-k+l-l/PG[Wn]nG[u])

=

/IHnl]/ (G[wntG[uJ) ,

and if we consider that here the fraction on the right hand side is I/G[h n ]' it follows that lim inf En+k n ....... oo

'

(lTp2-n-k+l-l/PG[WntG[U]) ;::: 1.

The proof of the upper estimate is completely symmetric if we use Theorem 10.2 and the polynomials R2n(X) = IHn- 1 (xW (I - x 2) there which are positive in (-1,1) with simple zeros at ±1 (d. (3.38)). Since the degree of H n in the proof can be smaller than n, say n - in with in -> 00, it also follows that the preceding argument actually holds for all k because eventually we shall have k ;::: -in. The uniformity of the convergence in k ;::: -K also follows from the proof. Finally, the last statement follows from Corollary 10.4 since the proof used only the existence of the polynomials guaranteed by this corollary (in Corollary 10.4 the degree of the polynomials H n can again be n - in where in -> 00).

•

We have already mentioned that the quantity E n ,2(W) gives the reciprocal of the leading coefficient of the n-th orthonormal polynomial with respect to the weight function W 2 . Thus, the case p = 2 is of special interest and is connected with many other problems in mathematics. Our next aim is to extend Theorem 13.1 in this case to more general weight functions u. This extension is connected to multipoint Pade approximation that we shall briefly discuss in Section 16.

Theorem 13.2 Let p = 2. Then with the assumptions of Theorem 13.1 the asymptotic relation (13.4) holds for every measurable u that is positive almost everywhere on [-1,1].

Section 13

93

Extremal problems with varying weights

We also add that in this case

(12

=

J1r/2,

so (13.4) has the form

Let us also mention that if u does not satisfy the the so called Szego condition (13.5)

1 1

logu(t) d t>-oo

,

then formula (13.4) gives only an upper estimate. Thus, in finer asymptotic problems we shall assume (13.5) exactly as is done in the classical case. Proof. We can assume that u satisfies the Szego condition (13.5). In fact, then the general case when this is not so follows by adding to u a positive band then letting b tend to zero. The case when u is continuous and positive follows from Theorem 13.1. The general case will be deduced with the aid of a theorem of G. Lopez [19]. Lopez proved the following: let

2n-j" V2*n =

II (x -

Zn,j)

j=1

be polynomials of degree 2n - in with in -+ 00 which are positive on (-1, 1), and let Jl be a measure which has positive (Radon-Nikodym) derivative almost everywehere in (-1,1). We set dJln = (V2'"n)-ldJl,.and assume that Jln is a finite measure on (-1, 1) for every n. Then we can form the orthonormal polynomials ,I: with respect to Jln:

1 1 -1

dJln = bl:,m.

Assume further, that if c)(z) = Z + y';2'-=1 is the conformal map that carries the complement of [-1, 1] into the unit circle, then for the zeros of the V2*n 's the condition 2n-j" 1 lim "(I-Ic)(znj)I) = 00 n-+oo L...i ' j=1

is satisfied. Under these conditions we have ([19, Theorem 9]) for n

-+ 00

(13.6) for every bounded f and every fixed k = 0, ±1, .... In other words, the measures converge weakly to the arcsine distribution in the stronger sense indicated.

94

Section 13

Applications

Let

,k be the leading coefficient of

,k' i.e.

As we have already mentioned, it is immediate from the orthogonality of the polynomials k that the extremal polynomial in the minimum problem (13.1) (with p = 2) the measure (V2*n)-ldJ.t is i.e.

J

2 . fin In r , 1 - 1 dJ.t PReII n V2"n A

=

J v;

* )

(

kI -:.:.= - d J.t. 2

V2"n

It also follows from the considerations of [19] and [21] that (see also [19, (43)]) (13.7) = (1 + o(I))y'1r2n+kG[V;nt/2G[jt'tl/2.

In fact, the corresponding result for the unit circle was proved in [21] under the assumptions that the zeros of certain polynomials (the analogues of V2n ) are real, but the proof works just as well for the general case if one uses the results of [19]. From here the transfer of the result to the real line to obtain (13.7) is just the standard technique (d. [33] or [19] and use also [19, Theorem 3]). After these preliminaries we turn to the proof of Theorem 13.2. By Theorem 10.2 there is a sequence {Hn-d of polynomials such that if x 2)- 1/ 4,

hn(x) = then hn(x)

1 and

lim G[hn] = 1.

n_oo

Actually, as we have remarked after the lemma, the degree of H n - 1 can be n-in with in 00. Furthermore, the construction of Lemma 9.1 shows that at least half of the zeros of the Hn are of distance Lncn- 1 from the interval [-1,1]' where L n 00 as n 00. Thus, if we set

V2';..(x) = IHn- 1(xW(1- x 2) and dJ.t(t) = (1 - t 2)1/2 U2(t) dt, then all the properties of V2*n mentioned above are satisfied (note that if dist(z, [-1, 1]) cLn/n, then 1-

ci

min(l, Ln/n)),

hence for the corresponding orthogonal polynomials hn 1 implies

s En+k,2 and so it follows from the formula

(13.8)

E n+k,2

Uipl/4

(

(v,* )1/2 2n

)

1

= -*--, In,n+k

k

'

we have (13.7). But

,

Section 14

Asymptotic properties of orthogonal polynomials with varying weights

95

the asymptotic relation (13.7) and

that

s1

n-oo

The proof of the lower estimate is similar if we use Theorem 10.1 instead Theorem 10.2. This completes the proof.

I

14

Asymptotic properties of orthogonal polynomials with varying weights

Let again {wn } be a sequence of weights on [-1, 1] as before, and u a measurable function satisfying the Szego condition (13.5). We denote by Pn,k(X) = 'Yn,k(X)X

k

+"',

'Yn,k

> 0,

the k-th orthonormal polynomial with respect to

Note the square in the weight. For the monic polynomials 1 qn,k := --Pn,k 'Yn,k

we have (14 .1)

J

2 ( n ) qn,kqn,kWn2n U2 -- E k,2 W U,

where Ek,2 is the extremal quantity discussed in Section 13. This means that the monic orthogonal polynomials are the optimal ones in the extremal L 2 problem of Section 13 for the case P = 2, and we also have (14.2)

'Yn,k =

1/Ek,2(wnu).

In this section we discuss asymptotics on Pn,k. For fixed weights (w n == 1) all these are classical, and some of the results below were proved by G. Lopez [19, 21] for varying case when the are reciprocals of polynomials. For simpler notation let Wn be the weight With some positive TJ > 0 we choose a stricly positive continuous function u· that coincides with u outside a set E" of measure smaller than TJ. By Theorem

96

Applications

Section 1"

10.2 (see also the remark after it) there is a sequence {Hn-d of polynomials of degree n - in, in -> 00, such that that they do not vanish on (-1,1) and if

s 1 and

then (14.3)

lim

n-oo

We also set (14.4)

= l.

x(x) = min{u*(x)/u(x),l}

which again belongs to the Szego class. By chosing 1] sufficiently small (and u" appropriately) the geometric mean of u* can be as close as we wish to that of u, hence the geometric means of XWn and Wn will also be close. Now with

(14.5)

*_ Wn -

we have

ep1/4

IHn-1 Iep 1/2'

s

ep(x) = 1 - x 2

s 1,

x E [-1,1]

and the geometric means of the three weights Wn, XWn and can differ by as small amount as we wish if we choose 1] and u* appropriately and n is sufficiently large. These properties and Theorem 13.2 on the asymptotic behavior of the leading coefficient of orthogonal polynomials with respect to varying weights easily imply the following: for every { we have polynomials H n of degree n - in with in -> 00 such that for some set E, of measure at most e we have the relations (14.6) = 1 + 0(1), and for every fixed k = 0, ±1," . and large n

(14.7) (14.8) and (14.9)

E;+k,2(XWn)

s

< =

s (1 +

where m denotes the m-th monic orthogonal polynomial corresponding to (W';)2. ' We have already mentioned that by [19, Theorem 9] the functions

Asymptotic properties of orthogonal polynomials with varying weights

Section 14

tend to

1 w·_.-

1r

97

1

v'i'"=t2

in the sense that for every bounded and measurable

f

(14.10) We start with a simple observation. We apply the parallelogram law

J

+

(Pn,n+k -

= '12

J(

J

Pn,n+k) 2(XWn) 2 + '12

+

J(

2

(XWn)2

* 2(XWn)2 , Pn,n+k)

and observe that the first term on the right is at most 1/2 by X 1, the second one is at most (1 + f)2/2 by (14.9) and (14.8) (see also (14.2)), while the second term on the left is at least as large as (1- f)2 by (14.7) and (14.2). Therefore, we can conclude that

(14.11) in particular,

f

(14.12)

lE;

(Pn,n+k -

12f,

where := [-1,1] \ E£ denotes the complement of E£ in [-1,1] (recall that X(x) = 1 for all x E Let now T be a measurable subset of [-1,1] not intersecting E£. We can easily get from (14.12), (14.6), (14.10) and Schwarz inequality that lim inf n_oo

lTf (Pn,n+k Wn)2 lTf w -

12f - 2JI2;,

which gives for e -+ 0 that for every measurable subset T of [-1, 1]

On applying this to the complement of T in [-1,1], it follows that here the liminf can be replaced by limit (recall, that the corresponding integral over [-1, 1] is 1). Since every bounded function can be uniformly approximated by linear combinations of characteristic functions of sets, we finally arrive at Theorem 14.1 Let {wn } be a sequence of weight functions on [-1,1] such that the corresponding extremal measures fJw n have support [-1,1], they are absolutely continuous there and if we write dfJwm (t) = V n (t) dt, then the functions V n

98

Section 1..

Applications

satisfy the conditions (13.2) and (13.3). Let furthermore u satisfy Szego's condition (13.5). If Pn,m are the orthonormal polynomials with respect to then for every fixed k = 0, ±I, ... and for every bounded and measurable f lim

n-+oo

J

=

f(Pn )

J

fw.

The result is also true for even weights {w n} satisfying on [0,1] the conditions of Theorem 10.3. For fixed weight, i.e. when W n == 1 this was proved in [33, Theorem 11.1], while, as we have seen and used above, for varying weights when is the reciprocal of a polynomial with zeros not too close to [-1,1], by Lopez [19, Theorem 9]. It would be equally easy to extend other results from [33] and [19] to the present case at least under Szeg8's condition on u. Probably the results also hold if we only assume that u(t) > 0 almost everywhere on [-1,1]. Instead of pursuing this direction further we seek stronger asymptotics on the orthogonal polynomials. As one can expect, We shall get strong asymptotics away from [-1,1], and on [-1,1] in L 2 norm. To this end we shall need the following corollary to Theorem 14.1. With the notations applied before the theorem we have (14.13)

as n -

J

00.

In fact, with the set E( used before we have

(Pn,n+k Wn -

+2

2

s 2 (f iE, (Pn,nHWn)2 +

f (Pn,nHWn JE;

1

+2 {

E,

-

lE;

and by the preceding theorem, (14.10), (14.12) and (14.6) we get the bound

!11' iE, r w + 24{ + o(1) for the right hand side, and this is easily seen to be smaller than 0 for large n (and small e). To formulate our next theorem we need a definition (see [48, (12.2.3)], where an additional factor 1/2 appears on the right which we put into our functions). If V is a nonnegative function on [-1, 1], then let fv(x) =

2.1

1

1l'

-1

log

-log V(x) x

e-

(1- e x

1-

2

) 1/2

2

where the integral is understood in principal value sense. It is easy to see that fv(cos (J) coincides for 0 0 11' with the trigonometric conjugate of log V( cos 0) (see (48, (12.2.3)1).

Section 14-

Asymptotic properties of orthogonal polynomials with varying weights

99

Theorem 14.2 Let W n satisfy the assumptions of Theorem 14.1, and suppose that u satisfies Szeg/J's condition (13.5). Then for fixed k = 0, ±1, ... the difference Pn,n+k(X

-{;h

(x )u(x) cos

((n + k+ 4) arccos X+ nrWn(X) + ru(x)

tends to zero in L2[-I, 1]. Proof. We continue to use the functions and notations from before Theorem 14.1. Recall that there we have chosen to an { > 0 a function u· and polynomials H n - 1 with properties (14.3)-(14.9). It follows from [48, Theorem 2.6, (2.6.2)] for p(x) = that cos 0)W: (cos 0)(sin 0)1/2 = (2/tr)1/25R { ei (n+k )8exp( irIHn_l! (cos 0» } for all large n (so that n - in < n + k). The last factor on the right is

so in view of (14.13) it is enough to prove that the difference of

and cos

((n + k +

+ nr Wn (cos 0) + ru(cosO)

tr] is as small as we wish for small f. Since the conjugate function of in log [aine] is 0 - (tr/2) for 0 E [O,tr] and (tr/2) + 0 for 8 E [-tr, 0], the last but one expression is actually cos ( (n + k +

+ r w: (cos 8) -

,

and so it is enough to show that the two functions

rw: (cosO) and

r w;:u( COB 8)

are close on a set of almost full measure on [0, tr] provided n is sufficiently large. But the difference of these two functions is and so recalling that rv(cos8) coincides with the trigonometric conjugate oflogV(cos8), it follows

100

Section 14

Applications

from the weak (1,1) property of the operator of trigonometric conjugation that the measure of the set

is at most

C TJ - 1

1 1r

=

CTJ-1 ( f1

J0 v"f"=X2

dx +

t'

J0

v"f"=X2

dX) .

Since for fixed TJ the first term on the right hand side tends to zero as n ---+ 00 (see Theorem 10.1) and the second term is as small as we like by appropriately choosing u", the proof is complete.

I

Finally, we prove a strong asymptotic formula for Pn,n+k away from [-1,1]. To this end we introduce the so called Szego function

Dv (z ) -exp

(/271 211"1 1 log V(x) 1

-1

Z-

x

dx

x2

)

provided log V(x)/v"f"=X2 is integrable, i.e. provided V satisfies the Szego condition (13.5). This form of the Szegd function appears in [22], and can be deduced from the more familiar one corresponding to the unit disk by the standard conformal mapping between C \ [-1, 1] and the unit disk. The Szego function for the unit disk is (see e.g. [48, Ch. 10])

lei < 1, This ih is the outer function associated with K 1 / 2 normalized by D(O) 0; in particular, DK is not zero in the unit disk, DK E H 2, and K(O) = IDK(ei6)1 2 almost everywhere, where DK (ei8) denotes the nontangential boundary limit of DK at the point ei8. If we set K(t) = V(cost), then Dv(z) = DK(e), = z-n-=1, where, as usual, we choose that branch ofthe square root that is positive for positive z. Hence, Dv is not zero in C\[-l, 1] and V(x) = IDv(xW for almost every x E [-1,1], where again the last quantity is a boundary limit. The function r that we used above gives the argument of DK on the lower part of the cut c \ [-1, 1] (see [48, (12.1.7)] and note that x = cosO, 0 $ 0 $ 11" on the lower part of the cut corresponds to the point ei8 under the mapping

e

e= z-n-=1): (14.15)

=

Section 14-

Asymptotic properties of orthogonal polynomials with varying weights

101

Theorem 14.3 With the assumptions of Theorem 14.1 we have

uniformly on compact subsets of C \ [-1, 1], where

We can put Theorem 14.3 into a somewhat different form (d. [43]). For simplicity let us assume that u is identically one (this can be attained in the most interesting cases by looking at w n u 1 / n instead of w n), in which case we will not need the Szego function in our asymptotic formula. Theorem 14.4 With the assumptions of Theorem 14.1 we have that the polynomials Pn.n+k(Z) are asymptotically equal to

+ v9'"=1)k+ 1/ 2 exp ( nFwn -

n / log Z

t dJ.lw n) (z2 - 1)-1/4

uniformly on compact subsets of C \ [-1,1], where J.lw n and FW n are the equilibrium measure and the equilibrium constants from Theorem A, Section 1. If u is not identically one, then the fixed multiplier

also appears on the right.

This form gives via standard arguments the following asymptotics for the zeros of the orthogonal polynomials: with the assumptions of Theorem 14.1 let V n be the normalized counting measure on the zeros of the orthogonal polynomials Pn,n+k' Then uniformly in the intervals I [-1,1]. Of course, to conclude this we do not need the full force of the strong asymptotic formula in the preceding theorem, n-th root asymptotics would suffice (see e.g. [47, Chapter 3]). That the two theorems Theorem 14.3 and 14.4 are equivalent can be easily seen from the fact, that the Szeg8 function associated with ../f=X'2 is 1/ 2

JZ2=1 (

)

,

and, by Theorem A in the introduction the function

(z + v9"=1) exp ( / log z t dJ.lwn(t) - Fwn)

102

Section 14

Applications

coincides with the outer function (on the domain C \ [-1,1]) associated with W n . Thus, it is enough to prove one of them, but before we do that let us utilize once more the preceding formula. By what we have said about the relation of rv and DK in (14.15) it follows from the just metioned identity that

where VWn(t) is the density of the equilibrium measure Jlw n' Since the argument of (z - t)-1 on the lower part of the cut of C \ [-1, 1] is 1C' for t > lRz and zero otherwise, it readily follows that

rWn(X) = 1C'

1 1

vwn(t)dt - arccos x,

and so we get the following variant of Theorem 14.2.

Theorem 14.5 With the assumptions of Theorem 14.1 for k = 0, ±1, ... the difference

-If

)u(x) cos

((k

arccos x + n1C'

1 1

vwn(t)dt + ru(x)

tends to zero in L 2[-1, 1].

Proof of Theorem 14.3. We use again (14.11). If we recall the properties of the Szego function this can be written with e = z - R=1, i.e. z = + lie) in the form

where

n

n f , and, as before, denote the orthonormal polynomials with respect to (see (14.5)). But here both functions under the integral sign belong to the Hardy space H 2 , therefore we obtain from Cauchy's formula, that uniformly on compact subset of the unit disk {e lieI < 1} the difference (14.16) tends to zero if n n f and e --+ O. By the choice of the functions H n and h n we have for

Section 15

Freud weights revisited

103

the identity

fh.. (e)/ DK: (e) = D(xh.. )(cos) (e)2 , and since the norm of 10g[X(cos t)h n ( cos t)] is as close to zero as we like by choosing f > 0 sufficiently small and n large (see Theorem 10.1) and of course selecting the function u* in (14.4) appropriately, the ratio on the left of the preceding formula will be as close to 1 as we like. Hence it is enough to examine the behavior of It is known that

(see [48, Theorem 2.6, (2.6.2)] where this formula appears for Izl = 1 - actually in the form of the first displayed equation in the proof of Theorem 14.2 - which clearly implies the same formula for all z because both sides are polynomials in z), hence it is left to show that here the second term is the dominant one. But that is easy: the ratio of the first and second terms on the right has absolute value 1 on the unit circle. Since the Szego function of a nonegative trigonometric polynomial of degree I is a polynomial of degree I ([48, Theorem 1.2.2]), it follows that the ratio in question has a zero at the origin at least of order 2(n + k) - 2(n - in), where n - in is the degree of Hn-l. Thus, by Schwarz's lemma the ratio of the first and second terms tends to zero uniformly on compact subsets of the unit disk as n - 00 because in - 00. This proves

From here we obtain the theorem if we use the aforementioned fact that the difference in (14.16) is as small as we like provided we choose c small and n sufficiently large (and u* in (14.4) appropriately), furthermore that with

the ratio DK.. (e)/Du.. (e) (which is nothing else than Dx(cost)(e)2) can also be as close to 1 as we like (see the definition (14.4) of X and recall that its geometric mean can be as small as we want).

I

15

Freud weights revisited

Let wa(x) = exp(-1alxla), 0: > 0 be the Freud weights we considered in Section 3, and consider the orthonormal polynomials with respect to

104

Section 15

Applications

We set for all n Z

E [-1,1].

Note that W n is defined on [-1,1]. Let Pn = 1 + n- 7 / 12 and

= wa(Pn z) = e--YaP"I3: la,

x E [-1,1].

It follows from the infite-finite range inequality (3.11) that

= (1 + 0(1))/

(15.1)

where E n ,2 is the extremal quantity from Section 13, (13.1) (see also the connection (14.2) in between this quantity and the leading coefficients of orthogonal polynomials). Here the support of the equilibrium measures associated with 1S

and not [-1,1] (see Section 3). But recall the discussion at the end of Section 9, where we proved that the estimates of Lemma 9.1 actually hold true on a larger range (see (9.26)), hence Corollary lOA is also true for symmetric weights which have support = [-en, en] with some en satisfying 1-n- r :$ en :$ 1 for some T > 0, and which otherwise satisfy the assumptions of of Theorem 10.3 on [O,en] rather than on [0,1]. In particularly, this is true for the weights we are considering now. But the proof of Theorem 13.1 was based on approximation like in Corollary lOA, hence Theorem 13.1 is also true for such weights, in 's, particular, for our On applying Theorem 13.1 we immediately get

= 1 + 0(1). Here 0'2 = J1r/2 and, as we have already seen in Section 3.1, (3.19),

which yields

Thus, we finally arrive at (15.2)

lim 'Yn(WOl)1r1/22-ne-n/an(n+1/2)/a = 1,

n-oo

which is the extension of (3.3) to all ex > O. Let now Z

E [-1,1],

Section 15

105

Freud weights revisited

and let Pn,k be the orthonormal polynomials associated with preceding section. We apply the parallelogram law

like in the

(Pn,n+k )2 Wn2n + 2 111 (Pn,n+k .. )2 2n -_ 2111 Wn, -1 -1

1:

and observe that the first term on the right is 1/2, the second one is at most 1/2 since

111

s

= 1,

Pn(Wa;

while the second term on the left is 1 + 0(1) because the ratio of the leading coefficients of and Pn,n+k tend to 1 (see (15.2) and Theorem 13.1, and also recall (14.2)). Thus, it follows that

1 1

.

(15.3)

hm n-+oo

-1

.. 2 2n_ (Pn ' n+k - Pn n+k) Wn - O. I

Using this relation instead of (14.13) everything that we have proven in the preceding section on W n can be carried over to the Freud polynomials. For example it follows from Theorem 14.5 that for fixed k = 0, ±1, ... the difference

-If h

n 1/2apn+k(Wa; n 1/ax) exp( -n,a Ixl a ) cos

((k

arccos x

+mr

1 1

va(t)dt

tends to zero in L 2[-I, 1], where Va is the Ullman distribution (3.4). We also mention that in [26] it was proved that pointwise asymptotics of this form are also valid (at least for a 3, see also [43] for another proof which covers every a> 1), i.e. the above difference tends to zero uniformly on compact subsets of (-1, 1) not just in L 2 norm. On the other hand, it follows from Theorem 14.4 and 1

+1

z-

Fw" - flOg -.!...-tdp.w" = log(z +

Jo

a- 1 zt2 dt - t2

vz

(d. the computation in Section 3) that n 1 / 2apn+k(Wa; n 1/ az) is asymptotically equal to 1 --(z + ..j;2-=-"i)n+k+l/2 exp ( n

..j2;

Jvz

zt

a- 1

2 -

t2

dt ) (z2 _ 1)-1/4

106

Section 16

Applications

uniformly on compact subsets of C \ [-1,1]. In fact, using (15.3), the proof of Theorem 14.3 can be copied to yield the preceding asymptotic relation from that of the one in Theorem 14.4. We could easily extend these asymptotic formulae for orthogonal polynomials with respect to a weight W 2 where W satisfies the conditions of Theorem 12.1. In fact, Theorem 10.3 can be applied in such case and otherwise the proof is just the same as before. Compare this with the results of [28] where similar results were proven under more restrictive conditions.

16

Multipoint Parle approximation

In this section we briefly discuss the problem of multipoint Pade approximation which is intimately connected to orthogonal polynomials with respect to varying weights. In fact, this area was the main motivation for A. A. Gonchar and G. Lopez [10] (see also [21]) for considering orthogonal polynomials with respect to varying weights and our discussion would not be complete if we did not touch this aspect of the theory. Historically orthogonal polynomials originated from continued fractions, and one of the classical results in the analytic theory of continued fractions is Markov's theorem to be discussed briefly below. A function of the form

(16.1)

f(z) = c +

J

dJ1-(x) x-z

is called a Markov function if J1- is a positive measure with compact support S(J1-) R i.e. Markov functions are Cauchy transforms of positive measures J1with compact support in R. For functions of type (16.1) A. Markov [Ma] proved that the continued fraction development

(16.2) of f at infinity converges locally uniformly in C \ I(S(J1-)), where I(S(J1-)) is the smallest interval containing S(J.l). In what follows we shall assume that the support of J1- lies in [-1,1]. It is well known that the n-th convergent is the [n - lin] Pade approximant to the function (1.1). Hence the convergents of (1.3) are rational interpolants with all interpolation points being identically infinity. Gonchar and Lopez considered rational interpolants with more general systems of interpolation points. For every n E R we select a set

An = {xn,o, ... , X n ,2n} of 2n + 1 interpolation points from C \ I(S(J1-)) , which are symmetric onto the

107

Multipoint Pade approximation

Section 16

real axis. The points need not to be distinct. We set

2n

(16.3)

Wn(z):=

IT

(z - Xn,j).

j=O

rt:n,;"t oo

The degree dn of Wn is equal to the number of finite points in An. Denote by 'Rn the set of all rational functions with complex coefficients with numerator and denominator degree at most n. By r n = rn(f,A n , · ) E'Rn we denote the rational function that interpolates the function f of type (16.1) in the 2n + 1 points of the set An = {xn,o, ... , Xn,2n}. If some of these points are identical, then the interpolation is understood in Hermite's sense. It is easy to see that this is equivalent to the assertion that the left-hand side of

f(z) - rn(f, An; z) = O(lzl-(2n+l)) wn(z)

as

Izl--+ 00,

is bounded at every finite point of An and at infinity it has the indicated behavior. We note that interpolation at infinity has not been excluded. It can be shown (see [10] or [47, Lemma 6.1.2]) that there exists a unique rational interpolant

of the above type, andp., satisfies the weighted orthogonality relation

f

ie dJl(x) Pn(X)X wn(x) = 0 for k = 0, ... , n - 1,

i.e. they are orthogonal polynomials of (exact) degree n with respect to the varying weights wn(x)-ldJl(x). Furthermore, the remainder term of the interpolant has the representation

(16.4) for all z E C \ [-1,1]. By homogeneity we can clearly assume that the Pn is the n'th orthonormal polynomial with respect to Wn(x) -1 dJl( x). Suppose that S(Jl) [-1,1]. Now this is a typical situation when the assumptions in the results of Section 10 hold true, at least if the points of An are not too close to [-1,1]. In fact, the function 1/lwn(z)1can be written as

1 U"'n(z) ----e Iwn(z)1 , where lin is the measure that has mass 1 at every point of An. Thus, if we set

108

Section 16

Applications

then the equilibrium measure jJWn corresponding to W n is nothing else than the balayage of vn/(2n + 1) out of C \ [-1, 1] onto [-1,1] plus

1- dn/(2n + 1) times the arcsine measure (equilibrium measure of [-1, 1]). It is easy to verify that if the points of An stay away from [-1, 1], then the collections of all such measures has the property, that the corresponding densities are equicontinuous in compact subsets of [-1,1] and they satisfy the conditions (9.1) and (9.2) with f3 = -1/2. Hence, the results of Section 14 can be applied provided the density u 2 of jJ is in the Szego class (see (13.5), and from the asymptotics there we can easily get strong asymptotics for the error away from [-1,1]. in view of (16.4) provided jJ is absolutely continuous and its density u satisfies the Szego condition (13.5). In fact, we know from Theorem 14.3 that Pn(z) asymptotically equals

+

_1)-1/4 (D u 2(z )) - 1

times the Szegd function (with respect to the domain C \ [-1, 1]) of W n . Let dn be the degree of w n . Then

and here the second factor Hn(z) on the right belongs to the Hardy space H 2(C \ [-1,1]), and the square of the Szego function associated with W n is just the outer function associated with H n . Recalling that the ratio of H n and that of its outer function is the Blaschke product (with respect to C \ [-1, 1]) associated with the zeros of Hn (note that there is no singular part in Hn ) , it follows that what remains in the ratio in front of the integral in (16.4) is

times the Blaschke product associated with the zeros of wn . The integral itself converges to .!. _1_ 1 dt _ 1 7r

J

-

JZ2"=1

by Theorem 14.1, so we have full description on how the remainder behaves away from [-1,1]:

where is the canonical conformal map of the complement C \ [-1,1] of [-1, 1] onto the exterior of the unit disk, and the product is taken for the zeros of W n , i.e. for the finite points in the system An.

Section 17

Concluding remarks

109

The results of G. Lopez ([19]-[21]) give a different asymptotics that supersede the above one in the sense that the points in An can approach the interval [-1, 1] so long as the sum E(I((xj,n)l- 1) tends to infinity. It is worth mentioning the connection of the above asymptotic relation with the problem of minimizing the norms of Blaschke products on compact sets. In fact, suppose that V is a compact subset of C \ [-1,1], and we want to construct good rational approximants to the Markov function f on V. By picking some sets An of 2n + 1 points the above discussed multipoint Pade approximants are certain one of the candidates. The problem is how to choose optimally the interpolation points in An so that the approximation be as good as possible. In view of the asymptotic relation given for the error, our problem is to minimize the uniform norm of the Blaschke product in the error on V in the presence of the weight function

IDu 2(Z)I·

A result of Parfenov [41] is relevant here, which asserts that if V is an ellipse with foci at ±1 (or the exterior of it), then the minimum behaves like (r + times the geometric mean of the weight, where r is the sum of the half axes of V. It is plausible that the points of An can be chosen so that this asymptotic is attained. It is an interesting problem to investigate the same question for other, less symmetric V's. Another interesting problem is how far the so obtained bound for the above rational approximation to f on V is from the best one.

17

Concluding remarks

As we have seen, the method of Section 2 gives good approximation for logarithmic potentials by logarithms of reciprocals of polynomials provided the generating measure has continuous density. The method was sufficiently strong to settle the approximation problem for weighted polynomials w n P n for a general class of weigths wand to considerably relax the conditions of [28] concerning approximation by weighted polynomials of the type W(a n .)Pn (-) . It is possible that finitely many logarithmic type singularities in the density (these arise for example at the origin if one considers w(x) = exp(-Ixl)) can be handled by appropriately adjusting the correction polynomials (like Sn-[n/A] in Section 2) to have appropriate order of interpolation at these 'bad' points. The situation is much worse if the infinite singularity is not of logarithmic type. For example, if w(x) = exp( -clx la) with 0 < Q' < 1, then the density of the extremal measure has a singularity of the form e- t a - 1 at the origin, and indeed, we know that in this case approximation is not possible. Internal zeros in the density function constitute another problem. We have seen in Example 4.5 that even a single zero may rule out the possibility of approximation in the sense of Theorem 4.2. On the other hand, Example 4.6 shows that in some cases approximation is possible even in the presence of an internal zero, and it seems to be a very delicate problem to clear the role of internal zeros on the approximation problem for given individual weights. The

110

Applications

Section 17

problem with internal zeros is that if the density function has a zero at Xo in the interior of Sw, then in general Xo will not belong to any SwA, A > 1 (c.f. the argument at the end of Section 6), and usually we need to apply the approximation technique to some w A instead of w in order to be able to handle the effect of the singularities in the density v that may appear around the endpoints. A typical example of this kind of difficulty is encountered if we consider the weight w( x) = eX' (note the positive coefficient in the exponent) considered on E := [-1,1]. It can be shown that Sw = [-1,1] the density v of jJw is given by

v(t) =

2t2

J1=t2' 1f 1 - t 2

which has a zero at the origin, and has a (1 - X 2)-1/2 type singularity at ±1 (see Section 11). If A > 1, then SwA will miss a neighborhood of O. We do not know e.g. iffor every function f E C[-1, 1] with f(±1) = 0 there is a sequence of polynomials Pn of degree at most n such that

uniformly on [-1,1]. Let us also mention that recently some efforts have been done to find a 'soft' approach to the approximation problem considered in this paper. In some restricted cases such an approach is possible, for example in [3] and [10] simple sign change counting was used to prove such theorems (this works for example if w(x) = e- X ' ) . The paper [18] should also be mentioned that contains a construction for related "one point" polynomials.

* * * Although I have not discussed the contents of this paper in details with D. S. Lubinsky and E. B. Saff, I would like to express my appreciation to them, because the present work was motivated by some of their results.

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Index approximation property 26 approximation property 21 balayage measure 28 Bernstein's formula 13, 20 Bernstein's problem 86 Bernstein-Walsh lemma 51 Blaschke product 108 Chebyshev polynomials 51 conformal map 93 energy integral 3 equilibrium or extremal measure 5 ofaset4 extremal or equilibrium measure 6 extremum problem 91 V-extremal problem 19 fast decreasing polynomials 79 Fekete or Leja points 26 Freud weights 1, 7 geometric mean 14 Hardy space 102 incomplete polynomials 23 infinite-finite range inequality 13 Jacobi weights 24 Laguerre weights 25 Lipshitz condition 53 logarithmic capacity 3 logarithmic energy 3 logarithmic potential 2 Maria's theorem 25 Pade approximation 106 Nikolskii-type inequality 19 orthogonal polynomials 11, 91

asymptotics for 99 leading coefficients of 11 recurrence coefficients of 11 outer function 100 parallelogram law 97 principle of domination 25 quasi-everywhere 3 rational interpolant 107 remainder term of 107 regular set 2 restricted support 22 strong asymptotics 10, 12 Szego condition 93 Szego function 100 trigonometric conjugate 98 Ullman distribution 8 weight function 3 admissible 3 weighted polynomials 1 Wiener's condition 2