Topics in Uniform Approximation of Continuous Functions [1st ed.] 9783030484118, 9783030484125

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Frontiers in Mathematics

Ileana Bucur Gavriil Paltineanu

Topics in Uniform

Approximation of Continuous Functions

Frontiers in Mathematics

Advisory Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) William Y. C. Chen (Nankai University, Tianjin) Benoît Perthame (Sorbonne Université, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (The University of New South Wales, Sydney) Wolfgang Sprößig (TU Bergakademie Freiberg) Cédric Villani (Institut Henri Poincaré, Paris) This series is designed to be a repository for up-to-date research results which have been prepared for a wider audience. Graduates and postgraduates as well as scientists will benefit from the latest developments at the research frontiers in mathematics and at the "frontiers" between mathematics and other fields like computer science, physics, biology, economics, finance, etc. All volumes are online available at SpringerLink.

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Ileana Bucur • Gavriil Paltineanu

Topics in Uniform Approximation of Continuous Functions

Ileana Bucur Department of Mathematics and Computer Science Technical University of Civil Engineering Bucharest, Romania

Gavriil Paltineanu Department of Mathematics and Computer Science Technical University of Civil Engineering Bucharest, Romania

ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-3-030-48411-8 ISBN 978-3-030-48412-5 (eBook) https://doi.org/10.1007/978-3-030-48412-5 © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com, by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Introduction

The uniform approximation of continuous functions by simpler functions has its origins in Weierstrass’ approximation theorems. Weierstrass published two results in 1885 when he was in his seventies where he shows that any continuous function on a real compact interval can be uniformly approximated by an algebraic polynomial, and, accordingly, any continuous periodic function of a period T = 2π can be uniformly approximated by a trigonometric polynomial. These results somewhat counterbalance his earlier, famous example of a continuous function on a real interval, which is not differentiable at any point on that interval. This example from 1861 shows that some continuous functions can be “very non-smooth,” while from Weierstrass approximation theorems it follows that a continuous function can be uniformly approximated by “very smooth” functions (i.e., polynomial functions). It is well known that an entire function (a function that allows a power series expansion) can be uniformly approximated on any interval from the domain of convergence of its power series by an algebraic polynomial (the corresponding Taylor polynomial). However, it is also known that only a restricted class of functions can be expanded in a power series; such a function must be, among other requirements, infinitely differentiable. Weierstrass’ theorem widely expands the class of functions that can be uniformly approximated by polynomials from entire functions to continuous functions. Weierstrass theorems are so important—both from a theoretical and practical point of view—that ever since they were published numerous mathematicians proposed various proofs for them. These proofs are extremely instructive, as their basic ideas can be successfully applied in other problems of mathematical analysis. In his excellent survey [32], Allan Pinkus classifies these proofs in three groups. The first group is composed of the proofs based on singular integrals; the original proof of Weierstrass, as well as the proofs given by Picard, Fejér, Landau and de la Vallée Poussin belong to this class. The second group contains the proofs based on the uniform approximation of some particular functions, such as the proofs given by Runge/Phragmén, Lebesgue, Mittag Leffler and Lerch. The proofs that do not belong to the first two groups form the third one; among these, we mention the proofs given by Bernstein, Volterra and Lerch.

v

vi

Introduction

All these proofs appeared prior to 1913. Half of a century later, in 1964, H. Kuhn published in Archiv der Mathematik an “elementary” proof of the theorem concerning the approximation by algebraic polynomials based on Bernoulli’s inequality. Allan Pinkus considers this the simplest and the most elegant proof. Kuhn’s proof is presented in Chap. 1, Sect. 1.1 of this book, along with the equivalence of the two Weierstrass’ approximation theorems: by algebraic and by trigonometric polynomials. Kuhn’s idea of using Bernoulli’s inequality was imitated by other mathematicians to the purpose of giving new “elementary” proofs for other approximating theorems. In this respect, we must specify that a “more elementary proof” does not necessarily mean an “easier proof”, but a proof using less “strong” mathematical results. Chapter 1 also presents various generalizations of Weierstrass’ theorems. Which leads to Sect. 1.2 in which we emphasize two elegant and consistent generalizations due to Korovkin and Bohman-Korovkin. A very consistent generalization was obtained by M.H. Stone in 1937; it was republished in 1948 with a simplified proof under the title “The Generalized Weierstrass Approximation Theorem,” in the journal Mathematics Magazine. Stone’s generalization follows two directions: on the one hand, passing from continuous functions on a closed real interval to functions which are continuous on an arbitrary compact space and, on the other hand, passing from the algebra of polynomials to a sub-algebra of C(K, R) satisfying certain conditions. This result, presently known as the Stone-Weierstrass theorem, is presented in Sect. 1.3. Like Weierstrass’ theorem, the Stone-Weierstrass theorem has many proofs, more or less elementary, from the proof given by Louis de Branges in 1959, who uses results from functional analysis (Hahn-Banach and Krein–Milman’s theorems), to the proof we present in Sect. 1.3 based on Brosowski and Deutsch’s paper [5], where they use Bernoulli’s inequality. It is a well-known fact that Stone-Weierstrass’ theorem is no more true in the case of complex functions, unless the subalgebra A of C(K, C) is self-adjoint, i.e., ∀ a ∈ A implies a¯ ∈ A. Erret Bishop [2] obtained a generalization of Stone-Weierstrass’ theorem to non-self-adjoint algebras. In Sect. 1.4. we present a very elegant proof of Bishop’s theorem, due to T.J. Ransford [35], who also uses Bernoulli’s inequality. Chapter 1 ends with Sect. 1.5, entirely based upon the results obtained by Paltineanu and Bucur [31]. The first part of this section introduces the notion of Uryson family. A theorem of density in the space of continuous functions defined on a compact space with values in the interval [0, 1] is then proved; this is a key theorem from which one can also deduce Stone-Weierstrass’ theorem, both in its algebraic and lattice variants as above. The second part of this section studies the subsets of C(X, [0, 1]) having the (V N) property (Von Neumann), and a Bishop type theorem for such sets is established. This theorem generalizes a result from 1992 due to Prolla the result which, in its turn, generalizes Von Neumann’s variant of Stone-Weierstrass’ theorem for continuous functions with range in [0,1]. We mention that both Prolla’s proof and our proof use Bernoulli’s inequality.

Introduction

vii

Following the generalization of the results from the first chapter, the second chapter presents various approximation theorems for continuous functions defined on a locally compact space. The non-compactness of the basic space gives rise to serious problems concerning the statement and the proof of such theorems. The problem of the kind of continuous functions we wish to deal with is taken into account: all of the continuous functions, only those which are continuous and bounded, those which vanish at infinity, those with compact support, etc. A natural frame for the study of such theorems proved to be that of the weighted spaces, introduced by Leopoldo Nachbin. A proper presentation of these spaces can be found in his paper [20]. The basic properties of the weighted spaces, as well as those of their duals, form the focus of Sects. 2.1 and 2.2. The second part of Chap. 2 presents some theorems of StoneWeierstrass type for a vector subspace or a convex sub-cone of a weighted space. The chapter ends with the generalization of these results for vector functions. In Chap. 3 we study the approximation of continuously differentiable functions. Bernstein’s theorem and a Stone-Weierstrass type theorem, due to L. Nachbin (see e.g. [26, pp. 104, 107]), are presented. Chapter 4, entirely based on the papers [14, 15, 21, 28–30], is devoted to the generalization of some of the previous results from the case of weighted spaces to the abstract case of the locally convex lattices of type (M). The real locally convex lattices of type (M) generalize the weighted spaces CV0 (X, R), while the complex locally convex lattices of type (M) generalize the weighted spaces CV0 (X, C). The idea of these generalizations belongs to Dan Vuza. It is generally known that a one-to-one correspondence exists between the closed subsets of a locally compact space X and the closed ideals of the weighted space CV0 (X, R). More precisely, −→ S ←− IS = {f ∈ CV0 (X, R); f |S = 0 } . This remark allows defining the notion of an antisymmetric ideal in a real locally convex lattice of type (M) by analogy with the notion of an antisymmetric set. Then two theorems of approximation for a vector subspace of a real locally convex lattice of type (M), in particular for a convex cone, are presented. The complex vector lattices, less known in the literature, were nevertheless studied by numerous mathematicians, such as: Romulus Cristescu, H.H. Schaefer, H.P. Lotz, W.A. Luxemburg and A.C. Zaanen, G. Mittelmayer and M. Wolff, W.J. De Schipper [39], Dan Vuza, etc. In Sect. 4.6, of preliminaries and notations, we used the paper. Dan Vuza: Elements of the theory of modules over ordered rings, Order Structures in Functional Analysis, Vol 2, Editura Academiei Române, Bucure¸sti, 1989, 175–283. We conclude Chap. 4 by introducing the antisymmetric ideal in a complex locally convex lattice of type (M), by generalizing the de Branges lemma and obtaining a theorem of approximation of the elements of such a lattice by elements belonging to one of its vector subspaces.

Contents

1

Approximation of Continuous Functions on Compact Spaces . .. . . . . . . . . . . . . . . 1.1 Weierstrass Approximation Theorems . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.2 Korovkin Type Theorems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.3 Stone-Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.4 Bishop Type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.5 Sets of Functions with VN Property (von Neumann Property) . . . . . . . . . . . . . .

1 1 7 19 30 36

2

Approximation of Continuous Functions on Locally Compact Spaces . . . . . . . 2.1 Weighted Spaces of Scalar Functions .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.2 Duality for Weighted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.3 Stone–Weierstrass Theorem in Weighted Spaces. . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.4 Stone–Weierstrass Theorem for Convex Cones in a Weighted Space . . . . . . . 2.5 Approximation of Vector Valued Functions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

47 47 51 65 78 82

3

Approximation of Continuously Differentiable Functions . . . . . .. . . . . . . . . . . . . . . 3.1 Preliminaries and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.2 Bernstein Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.3 Stone-Weierstrass Theorem for Continuously Differentiable Functions . . . .

91 91 94 97

4

Approximation Theorems in Locally Convex Lattices . . . . . . . . . . .. . . . . . . . . . . . . . . 4.1 Real Locally Convex Lattices: Preliminaries and Notations ... . . . . . . . . . . . . . . 4.2 Ideals in Weighted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.3 Antisymmetric Ideals of a Real Locally Convex Lattice of (M)-Type . . . . . . 4.4 An Approximation Theorem for a Vector Subspace of a Real Locally Convex Lattice of (M)-Type .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.5 An Approximation Theorem for a Convex Cone of a Real Locally Convex Lattice of (M)-Type .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.6 Complex Locally Convex Lattices: Preliminaries and Notations . . . . . . . . . . . 4.7 Antisymmetric Ideals in Complex Locally Convex Lattices of (M)-Type . . 4.8 The Generalization of de Brange Lemma for a Locally Convex Lattice of Type-(M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

101 101 103 105 112 114 121 125 129

ix

x

Contents

4.9 Approximation of the Elements of a Locally Convex Lattice of (M)-Type by Using One of Subspaces . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 133 Bibliography . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 137 Index . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 139

1

Approximation of Continuous Functions on Compact Spaces

1.1

Weierstrass Approximation Theorems

The first Weierstrass theorem shows that the set of algebraic polynomials is uniformly dense in the space of continuous real functions on a compact interval of R. The second Weierstrass theorem asserts that the set of trigonometric polynomials is dense, with respect to the same topology, in the space of all real 2π-periodic, continuous functions on R. In this section, we present the proof, given by H. Khun, of the first Weierstrass approximation theorem, followed by a proof of the equivalence of the above two assertions. Theorem 1.1.1 Let k be a natural number, k > 1, let a, b ∈ (0, 1) be such that a < and, for any natural number, let pn : [0, 1] → [0, 1] be the function given by

1 k

δ and n ∈ N is sufficiently large. For |x − xi | < δ it results n−1

|n (x)| ≤

ε ε = , 2·n 2

|gi+1 (x) − gi (x)| < n ·

i=1

and therefore   n−1     [gi+1 (x) − gi (x)] · pn (x − xi )]) ≤ |f (x) − g(x)| + |n (x)| f (x) − (g1 (x) +   i=1

< ε, ∀x ∈ [0, 1]. As the function Pε (x) = g1 (x) + function, the proof is finished.

n−1

i=1 [gi+1 (x)

− gi (x)] · pn (x − xi )] is a polynomial 

Remark 1.1.4 The Weierstrass theorem may be stated under the following form: Any real continuous function on the interval [a, b], a < b is the uniform limit of a sequence (Pn )n of polynomial functions on [a, b]. Indeed, the affine function ϕ : [0, 1] → [a, b] given by ϕ(t) = a + t · (b − a), t ∈ [0, 1] is a homeomorphism having as inverse the map ψ given by ψ(x) =

x−a , ∀x ∈ [a, b]. b−a

If f is a real continuous function on [a, b], then the function f ◦ ϕ defined on [0, 1] is continuous, and therefore there exists a sequence (pn )n of polynomial functions which converges uniformly to the function f ◦ ϕ on [0, 1]. Hence the sequence (Pn )n of polynomial functions on [a, b], given by Pn = pn ◦ ψ, converges uniformly to the function f = f ◦ ϕ ◦ ψ. We recall that a trigonometric polynomial is a function t : R → R of the form t (x) =

m k=0

(ak · cos k · x + bk · sin k · x) .

1.1 Weierstrass Approximation Theorems

Remark 1.1.5 For any polynomial function p(x) =

5

m

i=1 ci

· x i , x ∈ R, the functions

x → p(sin x), x → p(cos x) are trigonometric polynomials. We may express the functions x → cosn x, x → sinn x as trigonometric polynomials using the well-known identities: cos2 x =

1 1 + cos 2x 1 − cos 2x , sin2 x = , cos x · cos y = [cos(x + y) + cos(x − y)], 2 2 2

sin x · sin y =

1 1 [cos(x − y) − cos(x + y)], sin x · cos y = [sin(x + y) + sin(x − y)]. 2 2

Theorem 1.1.6 (Weierstrass [43]) Let f : R → R be a continuous 2π-periodic function. Then there exists a sequence (Tn )n of trigonometric polynomial which converges uniformly on R to f . Proof First we suppose that the function f is also symmetric, i.e., f (−t) = f (t) for all t ∈ R. Further we consider the continuous function ϕ : [−1, 1] → R given by ϕ(x) = f (arccos x), and the polynomial sequence of functions (Pn )n such that Pn is uniformly convergent to ϕ on the set [−1, 1]. Obviously the sequence of trigonometric polynomials (Tn )n , Tn = Pn ◦cos converges uniformly to the function f on the interval [0, π]. But the trigonometric polynomials Pn are all symmetric and so is the function f. Hence the sequence (Tn )n converges uniformly to f on [−π, π], and therefore on R, by periodicity. Generally, we show that the function S : R → R, defined by S(t) = f (t) · sin2 t, t ∈ R, is uniformly approximated by trigonometric polynomials on R. Indeed, using the above considerations we may consider two sequences (T1n )n , (T2n )n of trigonometric polynomials which are uniformly convergent on R to the symmetric functions g, h given by g(t) =

1 1 (f (t) + f (−t)), h(t) = (f (t) − f (−t)) · sin t. 2 2

6

1 Approximation of Continuous Functions on Compact Spaces

We have f (t) · sin2 t = g(t) · sin2 t + h(t) · sin t, and therefore the trigonometric sequence of polynomials (Tn )n defined by Tn (t) = sin2 t · T1n (t) + sin t · T2n (t), converges uniformly on R to the function S. We consider now a sequence of trigonometric polynomials (Qn )n which is uniformly convergent on R to the function C : R → R, given by C(t) = f

π 2

 − t · sin2 t.

Obviously the sequence of trigonometric polynomials (Rn )n , given by Rn (t) = Qn

π 2

−t



is uniformly convergent on R to the function t →C

π   − t = f (t) · sin2 − t = f (t) · cos2 t. 2 2

π

Hence the sequence (Tn + Rn )n is uniformly convergent on R to the function f .



Remark 1.1.7 The above theorem shows how the approximation theorem of continuous functions on compact intervals by algebraic polynomials implies the uniform approximation of 2π-periodic continuous functions on R by trigonometric polynomials. The converse implication is also true. Indeed, let f : [0, 1] → R be a continuous function and let f˜ be a continuous extension of f to the interval [−π, π] such that f˜(−π) = f˜(π) = 0. It is easy to verify that ⎧ f (0) ⎪ ⎨ π · (x + π), if x ∈ [−π, 0) f˜(t) = f (x) , if x ∈ [0, 1] ⎪ ⎩ f (1) 1−π · (x − π), if x ∈ (1, π]. Obviously, f˜ may be extended by periodicity to R, and for any ε > 0, we may consider a  trigonometric polynomial P , P (t) = m k=1 (ak · cos k · t + bk · sin k · t) such that   f(t) − P (t) < ε , ∀t ∈ R. 2

1.2 Korovkin Type Theorems

7

Using now Taylor uniform approximation of each function t → cos k · t, t → sin k · t, k = 1, 2, . . . , m by polynomial functions on the compact interval [−π, π], we may choose the algebraic polynomials Ck , Sk such that |ak · cos k · t − Ck (t)|
0 such that f (x ) − f (x ) < k if x − x < δk and then we consider, − + , qk,t defined for any k = 0 and t ∈ [0, 1] the polynomial functions of second degree qk,t by (x − t)2 (x − t)2 1 1 − + + f (t) + M · qk,t (t) = − + f (t) − M · , q (t) = k,t k k δk 2 δk 2 where M = 2 · f . One can see that − + qk,t (x) ≤ f (x) ≤ qk,t (x), ∀x ∈ [0, 1].

Indeed, if |t − x| < δk , then we have + f (x) − qk,t (x) = f (x) − f (t) −

(x − t)2 1 1 1 1 − 2 · f · < f (x) − f (t) − < − = 0. k δk 2 k k k

If |t − x| ≥ δk , then (x − t)2 1 1 1 f (x)−q + k,t (x) = f (x)−f (t)− −2· f · < f (x)−f (t)− −2 f < − < 0. k k k δk 2

1.2 Korovkin Type Theorems

9

Therefore we have + f (x) < qk,t (x), ∀x ∈ [0, 1], ∀t ∈ [0, 1].

Similarly it shows that − qk,t (x) < f (x), ∀x ∈ [0, 1], ∀t ∈ [0, 1].

Using now the positivity of Ln , we deduce the inequalities:     − + Ln qk,t ≤ Ln (f ) ≤ Ln qk,t + − −qk,t ≤ −f ≤ −qk,t .

Using the notation Mk =

M δk 2

and adding the previous inequalities, we obtain

    1   1 − + f (t) ·Ln (p0 )−Mk ·Ln (p1 − t · p0 )2 − −f (t)−Mk · (p1 − t · p0 )2 ≤ Ln (f )−f k k  ≤

   1   1 + f (t) · Ln (p0 ) + Mk · Ln (p1 − t · p0 )2 + − f (t) + Mk · (p1 − t · p0 )2 k k

(1.3)

Now, by hypotheses we have     lim Mk · Ln (p1 2 ) − p1 2  = lim Mk ·t · Ln (p1 ) − p1 = lim Mk ·t 2 · Ln (p0 ) − p0 = 0.

n→∞

n→∞

n→∞

and therefore   lim Mk · Ln ((p1 − t · p0 )2 ) − Mk · (p1 − t · p0 )2  = 0

n→∞

and lim |f (t)| · Ln (p0 ) − p0 ≤ lim M · Ln (p0 ) − p0 = 0.

n→∞

n→∞

Let now y ∈ [0, 1] be an arbitrary point. From the preceding considerations and the inequality (1.3), we get 1 − · (Ln (p0 ) − p0 ) (y) + f (t) · (Ln (p0 ) − p0 ) (y) − Mk k   ·Ln (p1 − t · p0 )2 (y) − Mk · (p1 − t · p0 )2 (y)

10

1 Approximation of Continuous Functions on Compact Spaces

1 · (Ln (p0 ) − p0 ) (y) + f (t) · (Ln (p0 ) − p0 ) (y) k   +Mk · Ln (p1 − t · p0 )2 (y) + Mk · (p1 − t · p0 )2 (y).

≤ Ln (f )(y) − f (y) ≤

On the other hand, we have   1   · (Ln (p0 ) − p0 ) (y) ≤ 1 · Ln (p0 ) − p0 , ∀k ∈ N∗ , ∀n ∈ N∗ k  k |f (t) · (Ln (p0 ) − p0 ) (y)| ≤ M · Ln (p0 ) − p0 , ∀t ∈ [0, 1], ∀n ∈ N∗ and         Mk · Ln (p1 − t · p0 )2 (y) ≤ Mk · Ln (p1 − t · p0 )2 − Mk · (p1 − t · p0 )2      + Mk · (p1 − t · p0 )2 (y)) , ∀n, k ∈ N∗ . We remark that, taking t = y then Mk · (p1 − t · p0 ) (y) = 0 and for any ε > 0, we can fix k sufficiently large such that ε 1 · Ln (p0 ) − p0 ≤ , for all n ∈ N∗ . k 4 And then, we take nε ∈ N∗ such that for all n ≥ nε , n ∈ N∗ , we have M · Ln (p0 ) − p0 ≤

 ε   ε    , Mk · Ln (p0 − t · p0 )2 − Mk · (p1 − t · p0 )2  < , 4 4

i.e., |(Ln (f ) − f ) (y)| < ε, for all n ∈ N∗ and all y ∈ [0, 1].



Definition 1.2.4 The linear, positive operator Bn , n ∈ N∗ , Bn : C ([0, 1]) → C ([0, 1]) given by Bn (f )(x) =

n k=0

f

  k · Cn k · x k · (1 − x)n−k , ∀f ∈ C([0, 1]) n

will be called Bernstein operator of order.

1.2 Korovkin Type Theorems

11

Corollary 1.2.5 (Bernstein, see e.g. [32, p. 30]) For any element f ∈ C([0, 1]) the sequence (Bn (f ))n is uniformly convergent to f on the interval [0, 1]. Since Bn (f ) is a polynomial function, we get a new proof of the first Weierstrass theorem. Proof We shall use the previous Korovkin theorem. It is sufficient to show that the sequences (Bn (pi ))n , i = 0, 1, 2 are uniformly convergent to p0 , p1 , p2 , respectively. (We use the notations from Theorem 1.2.3.) We have n

Bn (p0 ) =

Cnk · x k · (1 − x)n−k = [x + (1 − x)]n = 1 = p0 (x),

k=0

hence Bn (p0 ) − p0 = 0, ∀n ∈ N∗ . Let us denote pnk (x) = Cnk · x k · (1 − x)n−k , ∀ k ≤ n. Further we have n

k · pnk (x) = nx ·

k=0

n k=1

= nx ·

n−1

(n − 1)! · x k−1 · (1 − x)n−k (k − 1)!(n − k)!

C l n−1 · x l · (1 − x)n−1−l = nx ,

l=0

and therefore Bn (p1 )(x) =

n k · pnk (x) = x = p1 (x), Bn (p1 ) − p1 = 0, ∀n ∈ N∗ . n k=0

Analogously we have n

k · (k − 1) · pnk (x)

k=0

= n · (n − 1) · x 2 ·

n k=2

= n · (n − 1) · x 2 ·

n−2 l=0

(n − 2)! · x k−2 · (1 − x)n−k (k − 2)!(n − k)! Cn−2 l · x l · (1 − x)n−2−l = n · (n − 1)x 2

12

1 Approximation of Continuous Functions on Compact Spaces

and n

k 2 · pnk (x) = n · (n − 1) · x 2 + n · x; Bn (p2 )(x)

k=0

=

n 1 2 n · (n − 1) 2 x · k · pnk (x) = ·x + . 2 n n n2 k=0

Therefore we have     x − x 2   n(n − 1)x 2 + nx  1 2 Bn (p2 ) − p2 = sup  ≤ , − x  = sup 2 n n n x∈[0,1] x∈[0,1] from which it follows that lim Bn (p2 ) − p2 = 0.

n→∞

So, lim Bn (pi ) − pi = 0, i = 0, 1, 2, and the Korovkin theorem may be used. n→∞ Further we present Bohman–Korovkin theorem, a strong generalization of the above Korovkin theorem. For this we start with a Hausdorff compact space K, with the set C(K) of all real continuous functions on K endowed with the topology of uniform convergence. For any f ∈ C(K) we denote as usually by f the uniform norm of f .  Theorem 1.2.6 (Bohman–Korovkin, see e.g. [34, p. 324]) Let us suppose that K contains at least two points and let Ln : C(K) → C(K), k ∈ N∗ , be a sequence of positive linear operators. Let fi ∈ C(K), i ∈ 1, m have the following properties: (i) (Ln (fi ))n is uniformly convergent to fi for all i ∈ 1, m. (ii) There exists ai ∈ C(K), i = 1, m such that m i=0

ai (y) · fi (x) ≥ 0, ∀(x, y) ∈ K × K and

m

ai (y) · fi (x) = 0, iff x = y.

i=0

Then, for any f ∈ C(K) the sequence (Ln (f ))n is uniformly convergent to f .  Proof For any y ∈ K we denote Py = m i=1 ai (y) · fi and we remark that for any real number ε, ε > 0 there exists nε ∈ N such that   Ln (Py ) − Py  ≤ ε, ∀n ≥ nε , ∀y ∈ K.

1.2 Korovkin Type Theorems

13

Indeed, the functions ai , i ∈ 1, m are uniformly bounded on K, i.e., there exists M ∈ R+ ∗ such that ai ≤ M, ∀i ∈ 1, m. Further we have m m   Ln (Py ) − Py  ≤ ai (y) · (Ln (fi ) − fi ) ≤ M · Ln (fi ) − fi , i=1

i=1

ε and the last term is smaller then ε, if we choose nε ∈ N∗ such that Ln (fi ) − fi < m·M for all n ≥ nε and all i ∈ 1, m. Particularly, the sequence of continuous functions on K:

  y → Ln Py (y) : K → R is uniformly convergent to 0 on K. Indeed, we have             Ln Py (y) = Ln Py (y) − Py (y) ≤ Ln Py − Py  < ε, ∀n ≥ nε . We consider now two points y1 , y2 ∈ K, y1 = y2 and we denote by Q the element of C(K) given by Q = Py1 + Py2 . Since Py1 (x) = 0 if x = y1 and Py2 (x) = 0 if x = y2 , we deduce that Q is a strictly positive function on K and lim Ln (Q) − Q = 0.

n→∞

Let f be an arbitrary element of C(K) and let (y, x) → gy (x) = f (x) −

f (y) · Q(x) : K × K → R Q(y)

be a continuous function on K × K associated with f . Since this function vanishes on the diagonal  of K × K, we deduce that for any m ∈ N∗ , the set Um of K × K,     1   Um = (y, x) ∈ K × K; gy (x) < m is an open neighbourhood of . On the compact subset (K × K)\Um the above associated function gy is bounded, and the continuous function (y, x) → Py (x) is strictly positive. Let us denote   αm = inf Py (x); (y, x) ∈ (K × K) \Um

14

1 Approximation of Continuous Functions on Compact Spaces

and    βm = sup gy (x) ; (y, x) ∈ (K × K)\Um . Just from the definitions of Um , αm , βm , we have   gy (x) ≤ 1 + βm · Py (x) , ∀(y, x) ∈ K × K. m αm Since Ln is linear and positive, we get   Ln (gy ) ≤ 1 · Ln (1) + βm · Ln (Py ) , on K. m αm On the other hand the sequence (Ln (Q))n is uniformly convergent to Q on K and so, Q being strictly positive, the sequence (Ln (1))n is uniformly bounded, i.e., Ln (1.1) ≤ M  , for all n ∈ N. Hence for all n ∈ N, y ∈ K we have   Ln (gy )(y) ≤ 1 · M  + βm · Ln (Py )(y), m αm     Ln (f )(y) − f (y) · Ln (Q)(y) ≤ 1 · M  + βm · Ln (Py )(y),   m Q(y) αm         Ln (Q)(y)  f (y)    |Ln (f )(y) − f (y)| ≤ Ln (f )(y) − · Ln (Q)(y) + f (y) 1 −  Q(y) Q(y)     Ln (Q)(y) 1 βm ≤ · M + − 1 . · Ln (Py )(y) + f ·  m αm Q(y) Since Q > 0 and the sequence (Ln (Q))n is uniformly convergent to Q we deduce that    Ln (Q)   − 1 lim  = 0. n→∞  Q From the starting considerations the sequence, of functions y → Ln (Py )(y) is uniformly convergent to 0. The number M  is independent of m ∈ N. The numbers αm , βm depend  ε on m, but we fix ε > 0 and we choose a m sufficiently large such that M m < 3 . Then we ∗ take nε ∈ N such that      βm  Ln (Q)  ε  ε      α · Ln (Py )(y) < 3 for all y ∈ K , f ·  Q − 1 < 3 , ∀n ≥ nε , m

1.2 Korovkin Type Theorems

15

and in this way |Ln (f )(y) − f (y)| < ε, ∀y ∈ K, if n ≥ nε .



The following corollary shows that the Bohman–Korovkin theorem generalizes the old Korovkin theorem. Corollary 1.2.7 (Korovkin [13]) Let p0 , p1 , p2 be the polynomial functions on [0, 1] defined by p0 (x) = 1, p1 (x) = x, p2 (x) = x 2 , ∀x ∈ [0, 1], and let (Ln )n be a sequence of positive, linear operators on C ([0, 1]) such that lim Ln (pi ) − pi = 0, i = 1, 2, 3.

n→∞

Then for any f ∈ C ([0, 1]) the sequence (Ln (f ))n of functions of C ([0, 1]) converges uniformly to the function f , i.e., lim Ln (f ) − f = 0.

n→∞

Proof If we consider the compact interval K = [0, 1] and the functions a1 , a2 , a3 ∈ C ([0, 1]) given by a1 (y) = y 2 , a2 (y) = −2y , a3 (y) = 1 , y ∈ [0, 1], we remark that the function (y, x) → Py (x) : [0, 1] × [0, 1] → R+ defined by Py (x) =

3

ai (y) · pi (x) = (y − x)2

i=1

satisfies all requirements from Theorem 1.2.6 (Bohman–Korovkin’s theorem).



Furthermore, we show the second Weierstrass approximation theorem (by trigonometric polynomials) may also be derived from Bohman–Korovkin’s theorem. First, we introduce Cesàro-Fourier operators. For any continuous 2π-periodic function f on R, we denote by sn (f ) the partial Fourier sum of order n associated with function f : a0 + (ak · cos k · x + bk · sin k · x), 2 n

sn (f ) =

k=1

16

1 Approximation of Continuous Functions on Compact Spaces

where we have denoted  π  π 1 1 f (t) · cos k · tdt, bk = · f (t) · sin k · tdt, k ∈ 0, n. ak = · π −π π −π Replacing ak , bk in the above definition of Sn (f ) and using the well-known formula: sin(2n + 1) · 1 + cos a + cos 2 · a + . . . + cos n · a = 2 2 sin a2

a 2

we obtain 1 · π

sn (f ) =  ·



π −π

f (t)

 1 + cos t · cos x + . . . + cos n · t · cos n · x + sin t · sin x + . . . + sin n · t · sin n · x dt 2 

=

1 · π

=

1 2·π

π −π



 f (t) · π

−π

f (t) ·

 1 + cos(t − x) + . . . + cos n · (t − x) dt 2 sin(2 · n + 1) ·

t−x 2

sin t−x 2

dt.

Certainly Sn is a linear operator on the space C ([−π, π]), but it is not a positive one. By contrary, the following linear operators σn , so-called Cèsaro-Fourier sums, given by σn =

s0 + s1 + . . . + sn−1 n

are positive. Indeed, from the above expression of sn and using the formula: sin

sin2 n·a a 3·a (2n − 1) · a 2 + sin + . . . + sin = , 2 2 2 sin a2

we get 1 σn (f )(x) = 2n · π



π

−π

f (t) ·

n·(t −x) 2 dt. sin2 t −x 2

sin2

1.2 Korovkin Type Theorems

17

From the last formula, we deduce that Cesàro-Fourier sums are linear and positive operators on the space K ∗ ∼ = R/ ∼, where the equivalence relation on R is given by x ∼ y if x − y = 2nπ, n ∈ Z. This compact space K ∗ may be thought as the boundary of the unit disk in R2 : 

 (x, y) ∈ R2 ; x 2 + y 2 = 1 = {(cos θ, sin θ ); θ ∈ [−π, π] }

endowed with the trace topology or with the interval [−π, π) in which the base of neighbourhoods of −π is the family of subsets of the form:   

1 1  π − , π , n ∈ N∗ . −π, −π + n n Corollary 1.2.8 (Weierstrass [43]) Let f : R → R be a continuous 2π-periodic function. Then, for any ε ∈ R, ε > 0 there exists a trigonometric polynomial tε such that f − tε ≤ ε. We consider the compact space K ∗ = [−π, π). The 2π-periodic functions on R may be identified with their restrictions to K ∗ . On the space C(K ∗ ) we consider the linear and positive operators σn , n ∈ N, as before. For the functions f1 , f2 , f3 from K ∗ given by f1 (x) = 1, f2 (x) = cos x, f3 (x) = sin x, x ∈ K ∗ we have, by elementary calculus, σn (f1 ) = f1 , σn (f2 ) =

n−1 n−1 · f2 , σn (f3 ) = · f3 , ∀n ∈ N∗ , n n

and therefore lim σn (fi ) − fi = 0, i = 1, 2, 3.

n→∞

If we consider now the functions a1 , a2 , a3 from C(K ∗ ) given by a1 = f1 , a2 = −f2 , a3 = −f3 ,

18

1 Approximation of Continuous Functions on Compact Spaces

then for any x, y ∈ K ∗ we have Py (x) = a1 (y) · f1 (x) + a2 (y) · f2 (x) + a3 (y) · f3 (x) = 1 − cos(x − y) ≥ 0, and Py (x) = 0 iff x = y. Using now Theorem 1.2.6, we deduce that the sequence of trigonometric polynomials (σn (f ))n is uniformly convergent to f . The power of Bohman–Korovkin’ theorem is reflected in the following proof of Bernstein approximation theorem for continuous functions defined on compact subsets of Rn . Theorem 1.2.9 (Bernstein, see e.g. [32, p. 30]) Let K be the compact subset [0, 1]m of Rm and for any f ∈ C(K) let Bn (f ) ∈ C(K) given by Bn (f )(x1 , . . . , xm ) =

n

...

k1 =0

n

 f

km =0

km k1 ,..., n n

 · pnk1 (x1 ) · . . . · pnkm (xm ),

where we have put pnk (t) = Cnk · t k · (1 − t)n−k =

n! · t k · (1 − t)n−k , k ∈ 1, n. k!(n − k)!

Then the sequence of polynomial functions (Bn (f ))n is uniformly convergent to f on K. Proof We choose the following functions f1i , f2i , f3i , i ∈ 1, m from C(K) given by f1i (x1 , . . . , xm ) = 1, f2i (x1 , . . . , xm ) = xi , f3i (x1 , . . . , xm ) = xi 2 , i ∈ 1, m, and we consider the functions a1i , a2i , a3i , i ∈ 1, m from C(K) given by a1i (y1 , . . . , ym ) = yi 2 , a2i (y1 , . . . , ym ) = −2 · yi , a3i (y1 , . . . , ym ) = 1, i ∈ 1, m. Obviously, for any x = (x1 , . . . , xm ), y = (y1 , . . . , ym ) from K we have Py (x) =

m

f1i (x) · a1i (y) +

i=1

m i=1

f2i (x) · a2i (y) +

m

f3i (x) · a3i (y) =

i=1

m (yi − xi )2 , i=1

and the relations Py (x) ≥ 0 for all x, y ∈ K and Py (x) = 0 iff y = x are apparent. On the other hand, as in the proof of Corollary 1.2.5, we have   lim Bn (fj i ) − fj i  = 0, ∀j ∈ 1, 3, ∀i ∈ 1, 3.

n→∞

We finish the proof since the required conditions from Theorem 1.2.6 are fulfilled.



1.3 Stone-Weierstrass Theorem

1.3

19

Stone-Weierstrass Theorem

The most important generalization of Weierstrass approximation theorems was obtained by Marshall H. Stone in 1937 (see e.g. [40]). Eleven years after he gave a simplified proof of his result in [40]. The generalization goes in two directions: one consists in replacing the interval [0, 1] by an arbitrary Hausdorff compact space K and the other by changing the base of approximants, algebraic or trigonometric polynomials, into an arbitrary algebra of continuous functions on K. Nowadays, this generalization bears the name “StoneWeierstrass theorem” and it became an indispensable tool in analysis generally and a vital one in the study of continuous functions on a compact space. The following lemma, obtained by Bruno Brosowski and Frank Deutsch (see Lemma 2 in [5]) plays an essential role in the proof of Stone-Weierstrass theorem. Lemma 1.3.1 Let K be a Hausdorff compact space and let A be an algebra of real continuous functions on K which contains the constant functions and separates the points of K. Then, for any two disjoint and closed subsets F1 , F2 of K and any ε ∈ R, 0 < ε < 1 there exists a function a ∈ A such that (i) 0 ≤ a ≤ 1 (ii) a(x) < ε, ∀x ∈ F1 (iii) a(x) > ε, ∀x ∈ F2 . Proof Let x1 ∈ F1 , x2 ∈ F2 be arbitrarily chosen?. There exists a function ax1 ,x2 ∈ A with the properties: 0 ≤ ax1 ,x2 ≤ 1, ax1 ,x2 (x1 ) = 0, ax1,x2 (x2 ) > 0. Indeed, from the separating property of A, we may consider a function f ∈ A such that f (x1 ) = f (x2 ). Adding the constant function −f (x1 ), we obtain a function g ∈ A such that g(x1 ) = 0, g(x2 ) = 0. The function g 2 belongs to A and g 2 (x1 ) = 0, g 2 (x2 ) > 0, g 2 ≥ 0 on K. We take ax1 ,x2 =

g2 . g 2

20

1 Approximation of Continuous Functions on Compact Spaces

  The set [ax1 ,x2 > 0] = x ∈ K; ax1 ,x2 (x) > 0 is an open neighbourhood of x2 . Fixing x1 ∈ F1 , we have F2 =



[ax1 ,x2 > 0],

x2 ∈F2

and therefore, F2 being compact, we may choose a finite number of points x21 , . . . , x2n ∈ F2 such that F2 =

n 

[ax1,x2j > 0].

j =1

Obviously, the function ax1 ∈ A given by ax1 =

n 1 · ax1 ,x2j n j =1

vanishes at x1 , 0 ≤ ax1 ≤ 1 on K and it is strictly positive on F2 . Let us denote   δx1 = min ax1 (t); t ∈ F2 > 0.  We observe that the set ax1 < compact and

1 3

· δx1 is an open neighbourhood of x1 . Since F1 is

 

1 a x 1 < · δx 1 , F1 ⊂ 3 x1 ∈F1

we may choose a finite number of points, x1 , . . . , xm ∈ F1 such that F1 ⊂

 m

 1 a x i < · δx 1 . 3

i=1

As 0 < that

δxi 3

≤ 13 , we have 1 ≤

1 δx1


1 − ε, ∀x ∈ Fi , ai (x) < ε, ∀x ∈ K\Fi −1 . Hence for any ε ∈ (0, 1) there exists a function ai ∈ A such that 0 ≤ ai ≤ 1, 1Fi − ε ≤ ai ≤ 1Fi−1 + ε. Further we have n

n

n

n

i=1

i=1

i=1

i=1

2 2 2 2 1 1 1 1 un − ε = n · 1 Fi − ε = n · (1Fi − ε) ≤ n · ai ≤ n · (1Fi−1 + ε) 2 2 2 2 n

=

2 1 · 1Fi−1 + ε = vn + ε 2n i=1

un − ε < un ≤ f ≤ vn < vn + ε =

1 + un + ε, 2n

and therefore     2n   1 1  f − · a i  ≤ vn − un + 2 · ε < n + 2 · ε.  n 2 2  i=1  The proof is finished since ε > 0 and n ∈ N are arbitrary, and the function belongs to A.

 1 2n

·

2 n

i=1 ai

Remark 1.3.3 Stone-Weierstrass theorem extends the previous Weierstrass theorems (1.1.3 and 1.1.6) as well as the Berstein theorem for compacts in Rn (Theorem 1.2.9). Indeed, the set of polynomial functions on the interval [0, 1] as well as the set of trigonometric polynomials on the interval [−π, π] are both algebras of continuous

1.3 Stone-Weierstrass Theorem

23

functions containing the constant functions and separating the points of [0, 1] and [−π, π] for the trigonometric polynomials. The same argument justifies the density of the polynomial functions from Rn in the set of C(K) of all real continuous functions defined on a compact subset of Rn . Further we define the Uryson family of functions and also some of its applications will be presented. We remember that a topological space is called a normal space if for any non-empty closed and disjoint subsets F1 , F2 there exist two open disjoint subsets G1 , G2 such that F1 ⊂ G1 , F2 ⊂ G2 . It is well known that any Hausdorff compact space K is a normal space as well as any metrisable space. The famous Uryson theorem gives the following characterization of normal spaces: A topological space X is normal iff for any non-empty closed and disjoint subsets F1 , F2 there exists a continuous function f on X with values in the interval [0, 1] such that f (x) = 0, ∀x ∈ F1 ; f (x) = 1, ∀x ∈ F2 . This is the origin of Uryson family of continuous functions on a normal space introduced by I. Bucur [7] in order to unify the proofs of different density theorems. Definition 1.3.4 A family U of continuous real functions on a normal space X with values in [0,  1] is called Uryson family if for any closed, disjoint subsets F1 , F2 of X and any ε ∈ 0, 12 there exists a function u ∈ U such that u(x) ≤ ε, ∀x ∈ F1 ; u(x) ≥ 1 − ε, ∀x ∈ F2 . Remark 1.3.5 Let K be a Hausdorff compact space and let A be an algebra of real continuous functions on K containing the constant functions and separating the points of K. Then, the family A1 given by A1 = {a ∈ A; 0 ≤ a ≤ 1} is an Uryson family on K. The assertion follows from Lemma 1.3.1. Theorem 1.3.6 If U is an Uryson family of functions on a normal space X, then the convex covering of this family, co(U), is dense in the set C (X, [0, 1]) of all continuous functions on X with values in the interval [0, 1] if we endow C (X, [0, 1]) with the distance of uniform convergence on X.

24

1 Approximation of Continuous Functions on Compact Spaces

Proof For any continuous function f : X → [0, 1], and any n ∈ N, n ≥ 1, we denote    

   i i  Fi = f ≥ n := x ∈ X f (x) ≥ n , i ∈ 0, 1, 2, . . . , 2n . 2 2 A similar argument as in the proof of Theorem 1.3.2 shows that for any 0 < ε < exists ψi ∈ U such that     2n   1 1 f − < n + 2ε. · ψi    n 2  2  i=1 The proof is finished since the function

1 2n

2n

i=1 ψi

1 2

there



belongs to co (U).

Remark 1.3.7 Theorem 1.3.6 generalizes Stone Weierstrass theorem. Indeed, if A is an algebra of continuous functions on a Hausdorff compact space K which contains the constant functions and separates the points of K, then the set A1 , A1 = {a ∈ A; 0 ≤ a ≤ 1} = co (A1 ) , is an Uryson family of functions (see Remark 1.3.5). By Theorem 1.3.6, any continuous function f : K → [0, 1] is uniformly approximated by a sequence of A1 . But any real continuous function f : K → R is of the form: f = α1 · f1 − α2 · f2 , with α1 , α2 ∈ R+ , f1 , f2 ∈ A1 . Lemma 1.3.8 Let K be a compact Hausdorff space and let L be a vector lattice of continuous real functions on K which contains the constant functions and separates the points of K. Then the set: L1 = {a ∈ L; 0 ≤ a ≤ 1} , is an Uryson family on K. Proof Let x, y ∈ K, x = y be two points. Since L separates the points of K, there exists a function ψx,y ∈ L such that ψx,y (x) < ψx,y (y). Adding a constant function, eventually we may suppose ψx,y (x) < 0 < ψx,y (y). After multiplication with a positive constant

1.3 Stone-Weierstrass Theorem

25

function, the function ψx,y verifies the following inequalities: ψx,y (x) < 0 < ψx,y (y), ψx,y (y) > 1. Using the notations f ∨ g (resp. f ∧ g) for the supremum (resp. infimum) of the functions f and g we get the function ϕx,y :   ϕx,y = 0 ∨ ψx,y ∧ 1, which belongs to L, vanishing on an open neighbourhood of the point x and is equal to 1 on an open neighbourhood of the point y. Let now F1 , F2 be two closed and disjoint subsets of K, and for any x ∈ F1 , y ∈ F2 , let ϕx,y ∈ L be such that ϕx,y vanishes on an open neighbourhood U y (x) of x and it is constant equal 1 on an open neighbourhood U x (y) of y. If we fix x ∈ F1 , we have F2 ⊂ y∈F2 U x (y), and therefore F2 ⊂ ni=1 U x (yi ) for some choice of the points y1 , . . . , yn ∈ F2 . If we put ϕx =

n

∨ ϕx,yi , the function ϕx belongs to L, vanishes on an open

i=1

neighborhood U (x) of the point x and it is equal to 1 on F2 . With the above notations, we have F1 ⊂ x∈F1 U (x), and using the compactness of F1 we have F1 ⊂ m i=1 U (xi ) for m

some choice of the points x1 , . . . , xm ∈ F1 . It is now evident that the function ϕ = ∧ ϕxi belongs to L,ϕ = 0 on F1 and ϕ = 1 on F2 .

i=1



Theorem 1.3.9 (Stone-Weierstrass [7]) Let K be a compact Hausdorff space and let L be a vector lattice of continuous real functions on K which contains the constant functions and separates the points of K. Then L is uniformly dense in the space C(K) of all real continuous functions on K. Indeed, for any f ∈ C(K) we have f = f + − f − where f + = f ∨ 0 and f − = (−f ) ∨ f+ f− 0. Obviously the functions f , f belong to C(K, [0, 1]). On the other hand, since L1 = {a ∈ L; 0 ≤ a ≤ 1} is an Uryson family on K, just a convex one, we deduce, using f+ f− Theorem 1.3.6, that the functions f , f belong to the closure of L1 in C(K, [0, 1]). We have  f = f ·

f− f+ − f f



  ∈ · f · L1 − L1 ⊂ L, C(K) ⊂ L.

In 1959 Louis de Branges [4] give a new proof of the Stone-Weierstrass theorem concerning the algebras of continuous functions on Hausdorff, compact spaces, using two fundamental tools in functional analysis: Hahn-Banach and Krein-Milman theorems. Furthermore, we present this newer proof containing some ideas which may be useful in obtaining different Stone-Weierstrass type generalizations.

26

1 Approximation of Continuous Functions on Compact Spaces

In this section, K will be a Hausdorff compact space, C(K) the Banach space of all real continuous functions on K endowed with the uniform norm and M(K) will be the dual of (C(K), ), i.e., the set of all real linear continuous functionals μ on C(K). If for any two functions f, g ∈ C(K) we denote by [f, g] = {ϕ ∈ C(K); f (x) ≤ ϕ(x) ≤ g(x), x ∈ K} , then the closed unit ball in the Banach space (C(K), ) is the set [−1,1] where 1 is the constant function on K equal 1 at any point of K. M(K) endowed with the pointwise addition and multiplication with scalars from R is a linear space and just a Banach space if we endow M(K) with the dual norm: μ = sup {μ(f ); f ∈ [−1, 1]} . In fact, M(K) is a Banach lattice with respect to the order relation “≤” given by μ ≤ ν ⇔ μ(f ) ≤ ν(f ), ∀f ∈ C + (K), where C + (K) is the convex cone of all positive functions of C(K). If we denote by M + (K) the subset of M(K) given by   M + (K) = {μ ∈ M(K); μ ≥ 0} = μ ∈ M(K); μ(f ) ≥ 0, ∀f ∈ C + (K) , then for any μ ∈ M(K) there is the smallest element in M + (K) denoted by μ+ , such that   μ+ (f ) = sup μ(g); g ∈ C + (K), g ≤ f , for any f ∈ C + (K). Also there exists a smallest element in M + (K) denoted by μ− such that   μ− (f ) = sup −μ(g); g ∈ C + (K), g ≤ f , for any f ∈ C + (K). Moreover we have μ = μ+ − μ− and the measure μ+ + μ− is the modulus of μ denoted by |μ|, i.e., the smallest element ν in M + (K) such that |μ(f )| ≤ ν (|f |) , for all f ∈ C(K). In fact we have |μ| (f ) = μ+ (f ) + μ− (f ) = sup {μ(g); g ∈ C(K), |g| ≤ f } , for any f ∈ C + (K). We remember also that for any μ ∈ M + (K) we have μ = μ(1) and for any μ ∈ M(K) we have μ = |μ| = |μ| (1).

1.3 Stone-Weierstrass Theorem

27

Often, the elements of M(K) are called Radon measures. The reason for this denomination is the fact that for any positive element μ of M(K) there exists a positive measure λμ on B(K) the σ -algebra of Borel sets of K such that  f dλμ , ∀f ∈ C(K),

μ(f ) =

and λμ is uniquely determined by its regularity, i.e.,   λμ (A) = sup λμ (K); K − compact, K ⊂ A , for any A ∈ B(K). For an arbitrary μ ∈ M(K) there exists a sign measure λμ on B(K) such that  μ(f ) =

f dλμ , ∀f ∈ C(K),

namely, λμ = λμ+ − λμ− . In fact λμ+ (resp. λμ− ) is the positive (resp. negative) part of the measure λμ , i.e., there exists A ∈ B(K) such that λμ+ (A) = λμ+ (K), λμ− (K\A) = λμ− (K), and λ|μ| = λμ+ + λμ− . The support of an element μ ∈ M(K) is by definition the support of the associated measure λμ , i.e., the smallest closed subset F of K such that λ|μ| (F ) = λ|μ| (K). We denote the support of μ by Supp(μ). We remark that for any g ∈ L1 (λμ ) =   1 L λμ  the map, denoted by gμ , defined on C(K) by  gμ (f ) =

f · gdμ, ∀f ∈ C(K)

is linear and we have   gμ (f ) ≤ f ·



  |g| d λμ  = f ·



  |g| d λ|μ|  , ∀f ∈ C(K),

  ! i.e., this map belongs to M(K). One can show that gμ  = |g| dλ|μ| . ! Further from now on we shall freely use the notation μ(h) instead of hdλμ whenever the former expression makes sense. Also, for any subset E of C(K) we denote by E 0 the polar of E with respect to the duality C(K), M(K) namely: E 0 = {μ ∈ M(K); μ(f ) ≤ 1, f ∈ E} .

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If C is a convex cone in C(K) (resp. F a linear subspace of C(K)) we have C 0 = {μ ∈ M(K); μ(f ) ≤ 0, ∀f ∈ C } (resp. F0 = {μ ∈ M(K); μ(f ) = 0, ∀f ∈ F }). Lemma 1.3.10 (Louis de Branges [4]) Let F be a linear subspace of C(K) and let μ be an extreme point of the convex subset F 0 ∩ [−1, 1]0 of M(K). If g ∈ C(K) is such that gμ ∈ F 0 , then g is constant on Supp(μ). Proof Since 0 ∈ [−1, 1]0 and μ ≤ 1, we have necessarily μ = 1, otherwise μ is an interior point of the unit ball of F 0 , and, therefore, it is not an extreme point of this ball. Now, adding a positive constant function to g and multiplying this sum by a strictly positive number we can obtain a function h such that hμ ∈ F 0 , 0 ≤ h ≤ 1. We show that h is constant on Supp(μ). We consider the following measures μ1 , μ2 : μ1 =

μ + hμ μ − hμ , μ2 = . μ + hμ μ − hμ

Obviously, μ1 , μ2 belong to F 0 ∩ [−1, 1]0 , and we have μ + hμ = |μ| (|1 + h|) = |μ| (1 + h) μ − hμ = |μ| (|1 − h|) = |μ| (1 − h) μ + hμ + μ − hμ = |μ| (1 + h + 1 − h) = |μ| (2) = 2, μ + hμ μ − hμ + = 1, 2 2 μ + hμ μ − hμ · μ1 + · μ2 = μ. 2 2 Since μ is extremal, we have μ1 = μ2 = μ. Hence μ=

μ + hμ (1 + h)μ , = μ + hμ 1 + |μ| (h)

(1+ |μ| (h))μ = (1 + h)μ, h · μ = |μ| (h) · μ, h = |μ| (h) on Supp(μ).



Theorem 1.3.11 (Stone-Weierstrass [7]) Let K be a Hausdorff compact space and let A be an algebra of real continuous functions on K which contains the constant functions

1.3 Stone-Weierstrass Theorem

29

and separates the points of K. Then, A is dense in C(K) with respect to the uniform norm, i.e., A = C(K). Proof If A = C(K), then there exists g ∈ C(K)\A, and from Hahn-Banach theorem there exists μ ∈ M(K), μ = 1, such that μ(a) = 0, ∀a ∈ A and μ(g) = 0. Clearly, μ ∈ A0 ∩ [−1, 1]0 . Since the subset A0 ∩ [−1, 1]0 is compact and convex, from   " [−1, 1]0 =co Ext A0 ∩ [−1, 1]0 , hence Krein-Milman theorem, it results that A0 there exists ν ∈ Ext A0 ∩ [−1, 1]0 , ν = 0 such that ν(g) = 0. We remark that for any a ∈ A we have aν ∈ A0 and using de Branges Lemma we deduce that a is constant on Suppν. But A separates the points of K and therefore the support of ν is a singleton, i.e., Supp(ν) = {x0 } and so ν = ν(1) · εx0 . Since ν ∈ A0 and 1 ∈ A, we get ν(1) = 0, and so ν = 0 which contradicts the relation ν = 0. This contradiction comes  from the initial hypothesis A = C(K). Remark 1.3.12 The conclusion of Theorem 1.3.11 fails in the case of functions with complex values. Indeed, let D = {z ∈ C; |z| ≤ 1} and let H be the algebra of all continuous complex valued functions on D which are holomorphic in the interior of D. Obviously, H contains the constant functions and separates the points of D, since it contains all complex polynomials. Moreover it is closed with respect to the uniform convergence topology, but H = C(D, C), since the real continuous functions on D do not belong to H . Nevertheless, the conclusion in the Stone–Weierstrass theorem is still valid in the complex case if the algebra A is self-adjoint, i.e., for any element a ∈ A, its complex conjugate a¯ belongs to A. Remark 1.3.13 If A is a self-adjoint algebra, and we denote by Re(A) = {Re(a); a ∈ A}, then we have (i) Re(A) ⊂ A, (ii) Re(A) is an algebra, (iii) A = Re (A) + i · Re (A). Indeed, for any a ∈ A we have Re(a) = a+2 a¯ ∈ A and therefore Re (A) ⊂ A. Clearly Re(A) is a linear real vector space and Re(A) is an algebra, because if u1 , u2 ∈ Re(A) then u1 · u2 ∈ A and u1 · u2 = Re(u1 · u2 + i · 0) ∈ Re(A). Since for any f ∈ A, we have I m(f ) = Re(−i · f ) ∈ A we get that I m(f ) ∈ Re(A), and therefore A = Re(A) + i · Re(A). With these remarks we are able to establish the Stone-Weierstrass theorem for complex functions. Theorem 1.3.14 (Stone-Weierstrass [10]) Let K be a Hausdorff compact space and let A be an algebra (over the field of complex numbers) of continuous complex valued functions on K which is self-adjoint, contains the constant functions and separates the

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1 Approximation of Continuous Functions on Compact Spaces

points of K. Then A is dense in C(K, C)-the set of all continuous complex valued functions on K, with respect to the uniform norm, i.e., A = C(K, C). Proof By the above remarks, and by hypotheses, the real algebra Re(A) contains the constant functions and separates the points of K. Hence Re(A) is dense in C(K) = C(K, R). Further we have C(K, C) = C(K) + i · C(K) = Re(A) + i · Re(A) ⊂ Re(A) + i · Re(A) = A.

1.4



Bishop Type Theorems

Erett Bishop (see [2]) generalizes the Stone-Weierstrass theorem for the case of nonselfadjoint algebras of continuous complex valued functions. To present his results we need some preliminaries. Definition 1.4.1 Let K be a Hausdorff compact space and let A be an algebra of continuous complex valued functions on K. A subset S of K is called antisymmetric with respect to A or A-antisymmetric, if any element a ∈ A is constant on S if I m a ≡ 0 on S. Remark 1.4.2 A subset S ⊂ K is A-antisymmetric on S   iff any element a ∈ A is constant if a¯ |S ∈ A |S , i.e., there exists a  ∈ A such that a¯ S = a  |S or equivalently a S = a¯ |S , where for any complex number z we denote by z¯ its conjugate.   Indeed, if a S = a  |S with a  ∈ A, then I m(a + a  ) = 0 on S, and since S is Aantisymmetric, it follows that the function a + a  is constant on S. But Re(a) = Re(a  ) = 12 · Re(a + a  ) on S, hence Re(a) is constant on S. We have i · a |S = i · a  |S = (−i · a  ) |S ∈ A |S , and therefore the function Re(i · a) = −I m(a) is constant on S. If any element a ∈ A is constant on S if a |S ∈ A |S , then for any a ∈ A which is real on S we have a |S = a |S and therefore a is constant on S. Example 1.4.3 Let K be a compact subset of the complex plane C and let H be the algebra ◦

of all continuous complex functions on K which are holomorphic in K -the interior of K. ◦

Then any component E of K is H-antisymmetric. We remember that E is called component of a subset M of C if E is connected, E ⊂ M and there is no other connected subset of M larger than E. If M is an open subset of C,

1.4 Bishop Type Theorems

31

then any component E of M is open. Indeed, let x0 ∈ E and let B(x0 , r), r > 0 a ball included in the open set M. The sets B(x0 , r) and E being connected and E ∩B(x0 , r) = φ it follows that E ∪ B(x0 , r) is connected. Since we have E ⊂ E ∪ B(x0 , r) ⊂ M, ◦

we deduce E = E ∪ B(x0 , r) and hence x0 ∈ E . ◦ ◦ In the above case, if E ⊂ K is a component of K and if h ∈ H is such that h |E is real, we deduce that I mh = 0, and therefore the complex derivative of h on the connex, open set E is 0, i.e., h is constant on E. Coming back to the general case where K is an arbitrary Hausdorff compact space and A ⊂ C(K, C) is an arbitrary algebra, we shall denote by  the set of all antisymmetric with respect to A subsets of K. Just Definition 1.4.1 it follows: Remark 1.4.4 The family  has the following properties: (i) (ii) (iii) (iv) (v)

{x0 } ∈ , ∀x0 ∈ K.  If S1 , S2 ∈ , and S1 ∩ S2 = φ, then S1 ∪ S2 ∈  ∈ . If S ∈ , then its closure S ∈ . Any point x ∈ K belongs to a maximal A-antisymmetric subset denoted by Sx . K = ∪ {Sx ; x ∈ K}, where for any x, y ∈ K, x = y, we have either S1 ∩ S2 = φ, or Sx = Sy .

Further we shall use the following notations: For any closed subset F of K and any f ∈ C(X, C), we denote f |F the restriction of f to F and f F = sup { |f (x)| ; x ∈ F } = f |F . Also, for any non-empty subset M of C(X, C) we denote by dist (f |F , M |F ) the distance of f |F up to M |F , i.e.,   dist (f |F, M |F ) = inf f − m F ; m ∈ M . For F = K we have dist(f, M) = inf { f − m ; m ∈ M} . Lemma 1.4.5 The family of closed subsets F of Kwith the property: dist (f |F , M |F ) = dist (f, M) , has at least a minimal element with respect to the inclusion relation.

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1 Approximation of Continuous Functions on Compact Spaces

Proof Let F be the family of all closed subsets F of K such that dist (f |F , M |F ) = dist (f, M) , and let F0 be a subfamily of F totally ordered by inclusion. For any m ∈ M and any F ∈ F0 we denote F m = {x ∈ F ; |f (x) − m(x)| ≥ d} , d = dist (f, M) . Since for any F1 , F2 ∈ F0 we have F1 ⊂ F2 or F2 ⊂ F1 , we get F1 m ⊂ F2 m or F2 m ⊂ F1 m , and therefore the family (F m )F ∈F0 is totally ordered by inclusion. Just from the definition of F m and using Weierstrass theorem relative to the upper bound of a continuous function on a compact space, we deduce that F m = φ for all F ∈ F0 . " Since F m is compact for any F ∈ F0 we get {F m ; F ∈ F0 } = φ, and therefore ⎧ ⎨ ⎩

x∈

#

F ; |f (x) − m(x)| ≥ d

F ∈F0

⎫ ⎬ ⎭

=

#

F m = φ, m ∈ M.

F ∈F0

Denoting F0 = ∩ {F ; F ∈ F0 }, we get dist (f |F0 , M |F0 ) ≥ d i.e. dist (f |F0 , M |F0 ) = dist (f, M) , F0 ∈ F .



Theorem 1.4.6 (Machado [18]) Let A ⊂ C(K, C) be an algebra and let f be a continuous complex valued function on K. Then, there exists an antisymmetric S with respect to A such that dist (f, A) = dist (f |S, A |S ) . Proof From Lemma 1.4.5, we may choose a minimal element S with respect to the inclusion order relation in the set F of all closed subsets F of K such that f − A F = dist (f |F, A |F ) = dist (f, A) . We show that S is an antisymmetric subset with respect to A. For this, we consider an element a ∈ A such that the restriction of a to S is a real function. We want to show that a is constant on S.

1.4 Bishop Type Theorems

33

On the contrary case, we consider x0 , y0 ∈ S such that a(x0) = a(y0 ). We may suppose that a(x0 ) = 0 or a(y0) = 0 . Indeed, if a(x0) = 0 and a(y0) = 0 , then the function a  ∈ A: a  (x) = [a(x0) − a(x)] · a(x), ∀x ∈ K, satisfies the desired conditions because a  (x0 ) = 0 and a  (y0 ) = [a(x0) − a(y0)] · a(y0) = 0. 2 We observe that the function a0 ∈ A given by a0 = aa2 is positive, a0 (x0 ) = 0, 0 ≤ S a0 ≤ 1, and a0 = 1. Furthermore, we denote

  

   2 2 1 1 Y = a0 ≤ = x ∈ S; a0 (x) ≤ , Z = a0 ≥ = x ∈ S; a0 (x) ≥ . 3 3 3 3 Obviously, we have S = Y ∪ Z = (Y \Z) ∪ (Y ∩ Z) ∪ (Z\Y ), Y = S, Z = S, a0 (x)
, ∀x ∈ Z\Y. 3 3 3 3

Since S is an element minimal in , we get Y ∈ / , Z ∈ / . Hence there are two functions aY , aZ ∈ A such that f − aY < d, f − aZ < d. From Theorem 1.1.1, the sequence ϕn : [0, 1] → [0, 1] of functions given by 3n  ϕn (t) = 1 − t n  is uniformly convergent to 1 on the interval 0, 13 and is uniformly convergent to 0 on  the interval 23 , 1 , and consequently the sequence (an )n given by an (x) = ϕn [a0 (x)] is uniformly convergent to 1 on the set Y \Z and is uniformly convergent to 0 on the set Z\Y . If we denote bn = 1 − an , n ≥ 1, then (bn )n converges uniformly to 0 on Y \Z (or to 1 on Z\Y ). Hence the sequence (aY · an + aZ · bn )n converges uniformly to aY on Y \Z (or to aZ on Z\Y ) and therefore for n sufficiently large we have |f (x) − [an (x) · aY (x) + bn (x) · aZ (x)] | < d, ∀x ∈ (Y \Z) ∪ (Z\Y ). If x ∈ Y ∩ Z, then we have |f (x) − aY (x)| ≤ f − aY < d, and |f (x) − aZ (x)| ≤ f − aZ < d.

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1 Approximation of Continuous Functions on Compact Spaces

Hence if x ∈ Y ∩ Z, we have |f (x) − [an (x) · aY (x) + bn (x) · aZ (x)]| = |an (x) · [f (x) − aY (x)] + bn (x) · [f (x) − aZ (x)]|an (x) · |f (x) − aY (x)| + bn (x) · |f (x) − aZ (x)| < an (x) · d + bn (x) · d = d. From the preceding considerations, we have   f − [an · ay + bn · aZ ] < d, an · aY + bn · aZ ∈ A, S which contradicts the fact that S ∈ F . Hence S is A-antisymmetric.



Theorem 1.4.7 (Bishop [2]) Let A ⊂ C(K, C) be a subalgebra and let f be an element of C(K, C). Then we have f ∈ A, iff f |S ∈ A |S for any maximal antisymmetric set S with respect to A. Proof Using Theorem 1.4.6, there exists an antisymmetric set with respect to A, S0 such that dist (f, A) = dist (f |S0 , A |S0 ) . But by hypothesis dist (f |S0 , A |S0 ) = 0, and therefore dist (f, A) = 0, hence f ∈ A.  Remark 1.4.8 If A is a self-adjoint subalgebra of C(K, C) then for any antisymmetric set S with respect to A the functions a ∈ A are constant on S (we say that S is a constant set for A). The assertion follows directly from Remark 1.4.4. In fact, in this case, a set S is antisymmetric with respect to A iff it is a constant set for A. Giving an algebra A ⊂ C(K, C), for any x ∈ K we denote [x] = {y ∈ K; a(x) = a(y), ∀a ∈ A} . Obviously, [x] is the greatest constant set for A which contains x, and if x, y ∈ K we have either [x] = [y] or [x] ∩ [y] = φ. The family {[x]; x ∈ K} of closed subsets of K is a partition of K, i.e., K = ∪ { [x]; x ∈ K}.

1.4 Bishop Type Theorems

35

Corollary 1.4.9 If A is a self-adjoint subalgebra of C(K, C), then the following assertions are equivalent: (i) f ∈ A, (ii) f | [x] ∈ A | [x] , ∀x ∈ K, (iii)f | [x] ∈ A | [x] , ∀x ∈ K Indeed, the equivalence (i) ⇔ (ii) follows from Theorem 1.4.7 and the coincidence between the antisymmetric and the constant sets with respect to A. The equivalence (ii) ⇔ (iii) may be deduced from the equality A | [x] = A | [x] . Corollary 1.4.10 (Stone-Weierstrass [10]) If A is a self-adjoint subalgebra of C(K, C) which contains the constant functions and separates the points of K, then A is uniformly dense in C(K, C), i.e., A = C(K, C). Proof Since A separates the points of K, for any x ∈ K the set [x] is a singleton, i.e. [x] = {x} . On the other hand, since the constant functions belong to A, we have A | [x] = {ϕc ; c ∈ C} , ϕc : {x} → C, ϕc (x) = c . The proof is now finished if we apply the previous corollary.



As an application of Bishop’s theorem, we present the following Rudin’s result. Theorem 1.4.11 Let K be a compact subset of I n × C where I = [0, 1], such that for any t ∈ I n the section Kt = {z ∈ C; (t, z) ∈ K} does not separate C, that is the set C\Kt is connected. For any g ∈ C(K, C), we define the function gt : Kt → C by gt (z) = g(t, z). We suppose now that the previous function has the property that for any t ∈ I n the associated ◦

function gt is holomorphic on Kt -the interior of Kt . Then for any ε > 0 there exists a polynomial function P such that |g(t, z) − P (t, z)| < ε, ∀(t, z) ∈ K. Proof We denote by A the algebra of all polynomials defined on K with values in C. Since for any t1 , t2 ∈ I n the polynomial (t, z) → t − t1 2 separates the points t1 , t2 , we deduce that for any A-antisymmetric set S of K there exists t0 ∈ I n such that S ⊂ {t0 } × Kt0 .

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1 Approximation of Continuous Functions on Compact Spaces

To show that g ∈ A, it will be sufficient to prove that for any A-antisymmetric set S of K we have g |S ∈ A |S . For such set S let the point t0 ∈ I n such that S ⊂ {t0 }×Kt0 . By hypothesis, the function gt0 : Kt0 → C, given by gt0 (z) = g(t0 , z), is continuous on Kt0 and holomorphic on the interior of Kt0 . So, by Mergelyan theorem there exists a sequence of polynomials (Pn )n on C such that (Pn )n is uniformly convergent to gt0 on Kt0 . We observe now that the sequence (Qn )n of polynomials on K given by Qn (t, z) = Pn (z)  converges uniformly to the function g on the set S of K, i.e., g |S ∈ A |S .

1.5

Sets of Functions with VN Property (von Neumann Property)

Definition 1.5.1 We say that a family M of real functions defined on the arbitrary set X, containing the constant functions 0 and 1 is ϕ-convex, where ϕ ∈ M, if for any f, g ∈ M we have ϕ · f + (1 − ϕ) · g ∈ M. We remark that 1−ϕ ∈ M since 1−ϕ = ϕ ·0+(1 − ϕ)·1 and so M is also (1−ϕ)-convex. Moreover M is stable by multiplication with ϕ since ϕ · f = ϕ · f + (1 − ϕ) · 0. Lemma 1.5.2 If M is ϕ-convex, then ϕ n ∈ M for any n ∈ N∗ and M is ϕ n -convex. Hence M is (1 − ϕ n )-convex and also (1 − ϕ n )m -convex for any n, m ∈ N∗ . Proof Since ϕ · f ∈ M for all f ∈ M we get ϕ 2 · f, ϕ 3 · f , . . . , ϕ n · f ∈ M. Particularly, taking f = 1 we have ϕ n ∈ M. We show now inductively that M is ϕ n -convex for any n ∈ N∗ . We suppose that M is ϕ k -convex, i.e., ϕ k · f + (1 − ϕ k ) · g ∈ M, ∀f, g ∈ M. From hypothesis, we have ϕ k+1 · f + (1 − ϕ k+1 ) · g = ϕ · [ϕ k · f + (1 − ϕ k ) · g] + (1 − ϕ) · g ∈ M, for any f, g ∈ M, i.e., M is ϕ k+1 -convex.



Definition 1.5.3 We say that a subset M ⊂ C (X, [0, 1]) possesses the V N property if we have ϕ · f + (1 − ϕ) · g ∈ M, ∀ϕ, f, g ∈ M.

1.5 Sets of Functions with VN Property (von Neumann Property)

37

We observe that if the constant functions 0 and 1 belong to M, then M possesses the VN property iff M is ϕ-convex for any ϕ ∈ M. Remark 1.5.4 If M possesses the VN property and 0 ∈ M then ϕ · f ∈ M, ∀ϕ, f ∈ M and ϕ n ∈ M, ∀ϕ ∈ M, ∀n ∈ N∗ . If the constant functions 0 and 1 ∈ M, then 1 − ϕ ∈ M, ∀ϕ ∈ M and (1 − ϕ n )m ∈ M, ∀ϕ ∈ M for any n, m ∈ N∗ . The assertion follows from Lemma 1.5.2. Definition 1.5.5 Let K be a Hausdorff compact space and let M be a subset of continuous functions on K with values in the interval [0, 1] which contains the constant functions 0 and 1. A subset S ⊂ K is called antisymmetric with respect to M (M-antisymmetric) if any element ϕ ∈ M with the property: ϕ · f + (1 − ϕ) · g |S ∈ M |S, ∀f, g ∈ M, is constant on S. Furthermore, we denote by  the set of all antisymmetric subsets of K with respect to M. From Definition 1.5.5 we deduce the following remark. Remark 1.5.6 For any x ∈ K the set {x} belongs to , and there exists a maximal Mantisymmetric subset Sx such that x ∈ Sx . Moreover, for any x, y ∈ K we have either Sx ∩ Sy = φ or Sx = Sy and K = ∪ {Sx ; x ∈ K}. Theorem 1.5.7 Let M ⊂ C(K, [0, 1]) be such that the constant functions 0, 1 ∈ M and let f ∈ C(K, [0, 1]). Then, there exists a subset S ⊂ Kantisymmetric with respect to M such that dist (f, M) = dist (f |S , M |S ). Proof We denote by F the family of all closed subsets F ⊂ K with the property: dist (f |F, M |F ) = dist (f, M) = d. By Lemma 1.4.5, F has minimal elements with respect to inclusion as order relation. To prove the theorem, it will be sufficient to show that a minimal element S of F is antisymmetric with respect to M. We suppose the contrary and let ϕ ∈ M be such that ϕ · f + (1 − ϕ) · g |S ∈ M |S, ∀f, g ∈ M,

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1 Approximation of Continuous Functions on Compact Spaces

but ϕ is not constant on S. We choose y, z ∈ S and a, b ∈ [0, 1] such that 0 ≤ ϕ(y) < a < b < ϕ(z) ≤ 1. We may suppose that 2 · a < b. Indeed, since such that  a n b


− ≥ > >1; 2< < . a a b 2·a b b a So there exists k ∈ N such that 1 1 1