A continuous function with universal Fourier series on a given closed set of Lebesgue measure zero

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ScienceDirect Journal of Approximation Theory 252 (2020) 105361 www.elsevier.com/locate/jat

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A continuous function with universal Fourier series on a given closed set of Lebesgue measure zero S. Khrushchev New School of Economics, Satbayev University, 22a Satpaev str., 050013 Almaty, Kazakhstan Received 4 June 2019; received in revised form 1 December 2019; accepted 29 December 2019 Available online 3 January 2020 Communicated by P. Nevai

Abstract Given a closed set E of Lebesgue measure zero on the unit circle T there is a continuous function f on T such that for every continuous function g on E there is a subsequence of partial Fourier sums Sn+ ( f, ζ ) =

n ∑

fˆ(k)ζ k

k=0

of f , which converges to g uniformly on E. This result completes one result in a recent paper by C. Papachristodoulos and M. Papadimitrakis (2019), see Papachristodoulos and Papadimitrakis (2019). They proved that for a classical one third Cantor set C there is no universal function in the disk algebra. They also proved that for a symmetric Cantor set C ∗ on T there is no universal continuous function for the classical symmetric Fourier sums. See also [2]. c 2019 Elsevier Inc. All rights reserved. ⃝ MSC: 30D55; 42A20; 42B30; 46A22; 46E15; 46E35; 46J15 Keywords: Fourier series; Continuous functions; Universal Fourier series; Convergence of Fourier series

This paper was stimulated by [1], where my old results [5] were used to relate the problem of simultaneous approximation by polynomials to universal Fourier series in different spaces. Its purpose is to separate the arguments of [5] from the arguments of universal power series theory. For the theory of Hardy spaces see [3,4]. E-mail address: [email protected]. https://doi.org/10.1016/j.jat.2019.105361 c 2019 Elsevier Inc. All rights reserved. 0021-9045/⃝

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1. Simultaneous approximation Let X be a Banach space of analytic functions in the open unit disk D, which includes the set P of all polynomials in z, and E be a closed set on the unit circle T. We assume that for every r , 0 ≤ r < 1 there is a constant Cr > 0 such that for every f in X max | f (z)| ≤ Cr · ∥ f ∥ X .

(1)

|z|≤r

Condition (1) implies that any sequence of polynomials converging to an element f in X converges to f as to an analytic function locally uniformly in D. The second assumption on X is that the linear space P is dense in X : Clos X (P) = X.

(2)

Definition 1.1. We say that a pair (E, X ) satisfies S A-property if for any continuous function g on E and any analytic function f in X there is a sequence of polynomials pn (z) such that lim pn = g uniformly on E and lim pn = f in X .

n→∞

n→∞

For a closed subset E of the unit circle T = {z : |z| = 1} we denote by C(E) the Banach algebra of all continuous functions f on E equipped with the standard norm ∥ f ∥ E = sup | f (ζ )|. ζ ∈E

By M. Riesz Theorem the dual space for C(E) is the Banach space M(E) of all Borel measures on E. The duality between C(E) and M(E) is given by the formula ∫ ⟨g, µ⟩ E = g(ζ )dµ(ζ ). E

By (2) every continuous linear functional Φ on X is uniquely defined by its Fourier coefficients: Φ(z n ) = ⟨z n , Φ⟩ X , n = 0, 1, . . . . The duality in the direct product C(E) × X of Banach spaces C(E) and X is defined by the formula ∫ ⟨(g, f ), (µ, Φ)⟩ = g(ζ )dµ(ζ ) + Φ( f ). E

It is clear that the pair (E, X ) satisfies S A-property if and only if the linear span of the diagonal D = {(z n , z n ) : n = 0, 1, . . .} is dense in C(E) × X . By duality this happens if and only if ∫ 0= ζ n dµ(ζ ) + Φ(z n ) for n = 0, 1, . . . ⇒ µ = 0 and Φ = 0.

(3)

E

In [5] the problem of simultaneous approximation was completely solved for a wide class of pairs (E, X ). However, one simple case of simultaneous approximation important for this paper was not included in [5]. Let m be the Lebesgue measure on T. Every function f integrable with respect to dm determines a unique sequence of Fourier coefficients: ∫ n ˆ f (n) = ζ f (ζ ) dm(ζ ), T

where n is an integer. We denote the set of all integers by Z.

S. Khrushchev / Journal of Approximation Theory 252 (2020) 105361

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The closure of the set P of all polynomials in z in C(T) is called the disk algebra C A (T). By the maximum principle each function f in C A (T) extends analytically to the open unit disk D: f (z) =

∞ ∑

fˆ(k)z k , |z| < 1.

k=0

Let B = { f ∈ C(T) : fˆ(n) = 0, for n ≥ 0}. By M. Riesz theorem the dual space of C(T) is the space M(T) of all finite Borel measures µ on T, where the duality is given by the formula ∫ ⟨ f, µ⟩ = f (ζ¯ )dµ(ζ ). T

The annihilator of B in this duality consists of measures µ satisfying ∫ −n 0 = ⟨ζ , µ⟩ = ζ n dµ(ζ ) = µ(−n), ˆ n = 1, 2, . . . . T

By F. and M. Riesz Theorem such measures are absolutely continuous dµ = hdm the density h being the boundary values of a function from the Hardy class H 1 . It follows that the dual space of C(T)/B is H 1 . If Φ is a functional in (C(T)/B)∗ = H 1 then ∫ ζ n dµ(ζ ) + Φ(z n ) for n = 0, 1, . . . ⇒ dµ + hdm = r dm, r ∈ H 1 . 0= E

It follows that dµ is absolutely continuous and its Lebesgue derivative coincides almost everywhere with the function r − h, which is in H 1 . If m(E) = 0 this implies that µ = 0. But then Φ = 0 as well. Corollary 1.1. For every closed subset E of T with zero Lebesgue measure the pair (E, C(T)/B) satisfies S A-property. 2. Universal Fourier series In this section we assume that the shift operator f −→ z · f is bounded on X . Theorem 2.1. If (E, X ) satisfies the S A-property then there is a function f in X such that partial sums of its Taylor’s series Sn+ ( f, z) =

n ∑

fˆ(k)z k

(4)

k=0

approximate any continuous function g on E. Proof. It is not possible that E = T since otherwise z¯ cannot be uniformly on T approximated by polynomials. Hence E ̸= T. It follows that polynomials are dense in C(E). Since f −→ z · f is bounded on X there is a constant C > 0 such that ∥z n · f ∥ X ≤ C n ∥ f ∥ X for n = 0, 1, 2 . . . .

(5)

We chose a sequence of polynomials {gk }k≥1 which make a dense set in C(E) and each member of this sequence occurs in the sequence infinitely many times. Applying the S A-property

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to the pair of functions (g1 , 0) we find a polynomial P1 (z) of degree n 1 − 1 such that max |g1 (z) − P1 (z)| < 2−1 and ∥P1 ∥ X < 2−1 . z∈E

Applying the S A-property to the pair of functions ((g2 − P1 ) · z −n 1 , 0), we find a polynomial P2 such that max |g2 (z) − P1 (z) − z n 1 P2 (z)| < 2−2 and ∥z n 1 P2 ∥ X ≤ C n 1 ∥P2 ∥ X < 2−2 . z∈E

Let n 2 − 1 be the degree of the polynomial P1 (z) + z n 1 P2 (z). Applying the S A-property to the pair of functions ( ) (g3 − P1 (z) − z n 1 P2 (z))z −n 2 , 0 , we find a polynomial P3 (z) such that max |g3 (z) − P1 (z) − z n 1 P2 (z) − z n 2 P3 (z)| < 2−3 and ∥z n 3 P3 ∥ X < 2−3 . z∈E

We continue by induction and obtain the following power series f (z) = P1 (z) + z n 1 P2 (z) + z n 2 P3 (z) + z n 3 P4 (z) + · · · + z n k Pk+1 (z) + · · · which converges to f in X . Its partial sums P1 (z) + z n 1 P2 (z) + z n 2 P3 (z) + z n 3 P4 (z) + · · · + z n k Pk+1 (z) approximate gk (z) uniformly on E with mistake not exceeding 2−k . Since each gk occurs in the sequence {gk }k≥1 infinitely often, the result follows. □ To complete the proof of the result stated in the Abstract we must show that the shift operator f −→ z · f is bounded on C(T)/B. Notice that every factor class f + B is determined by the power series f + (z) =

+∞ ∑

fˆ(k)z k

k=0

and the norm of f in C(T)/B is defined by ∥ f ∥ = inf{∥h∥C(T) : h + = f + }. The function f + is in the Hardy class H 2 . For z ∈ T we have ( ) ˆ z f + (z) = zh(z) − h(−1) . +

Since ˆ ˆ sup |zh(z) − h(−1)| = sup |h(z)| + |h(−1)| ≤ 2 sup |h(z)|, z∈T

z∈T

z∈T

we obtain that the shift operator is bounded with C = 2. Remark. A continuous function f in C(T) is called uniformly universal on a closed subset E of T if for every continuous function g on E there is a strictly increasing sequence of positive integers (n k )k≥1 such that the partial sums Sn k ( f, ζ ) =

nk ∑ j=−n k

fˆ( j)ζ j

(6)

S. Khrushchev / Journal of Approximation Theory 252 (2020) 105361

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converge uniformly on E to g. In [7] the class of all continuous functions f on E is denoted by U (E). Following [7], we identify points of T with points of the closed interval [−π, π] on the line via the mapping θ → eiθ . Let C be the one third Cantor set in [0, π] and C ∗ = C ∪(−C) be the symmetric Cantor set in [−π, π]. Recall that C A (T) is the disk algebra. Corollary 4.1 of [7] claims that (i) C A (T) ∩ U (C) = ∅, (ii) C(T) ∩ U (C ∗ ) = ∅. The result stated in the Abstract shows that (C(T)/B) ∩ U (C) ̸= ∅. The methods used in this paper cannot prove that C A (T) ∩ U (C) ̸= ∅ since the disk algebra does not satisfy the S A-property for any non-empty set E. One cannot approximate the identity function 1 on E in C(E) and the identity zero function 0 in C A (T) by one sequence of polynomials ( pn ). Indeed, by the maximum principle sup | pn (z) − 0| = sup | pn (z) − 0|, |z|≤1

|z|=1

which shows that convergence of pn to zero in C A (T) implies the convergence of pn to zero in C(T) and hence convergence to zero in C(E). This method cannot also lead to C(T) ∩ U (C ∗ ) ̸= ∅. The reason is that it does not allow one to control the partial sums (6). Instead, it allows one to control only the partial sums (4). I am grateful to a reviewer of this paper who turned my attention to interesting paper [7]. 1/ p

3. Besov spaces B p

The method of simultaneous approximation cannot be used directly to extend the results of existence of universal series in the disk algebra for any given closed subset E of T of zero Lebesgue measure. An obvious obstacle is that if polynomials approximate the zero function in A then they approximate zero in C(E). However, universal series exist for some closed subsets of T even in spaces of relatively smooth functions. There are many equivalent descriptions of 1/ p Besov spaces B p , some of which can be found in [6]. For 1 < p < +∞ the Besov space 1/ p B p is a subset of L p (T). The Riesz projection P+ , ∑ P+ f = fˆ(k)z k k≥0 1/ p

1/ p

is bounded on B p . The space P+ B p is the set of analytic functions f in D satisfying ∫ ∫ p | f ′ (z)| (1 − |z|) p−2 d xdy < +∞. D 1/ p

−1/ p

The adjoint space for P+ B p is P+ B p′ , where p ′ = p/( p − 1). An analytic function f is −1/ p in P+ B p′ if and only if ∫ ∫ ′ ′ | f (z)| p (1 − |z|) p −2 d xdy < +∞. D

The following Theorem was proved in [6], see Theorem 3.16. Theorem 3.1.

Let 1 < p ′ < +∞ and µ ∈ M(T). Then −1/ p

P+ µ ∈ B p ′

−1/ p

⇒ µ ∈ B p′

.

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S. Khrushchev / Journal of Approximation Theory 252 (2020) 105361 1/ p

The Besov space B p with the norm ( )1/ p ∫ π ∫ π p i(s+t) is p −2 | f (e ) − f (e )| ∥ f ∥ B 1/ p = ∥ f ∥ L p + t p

−π

−π

determines the capacity, see [6], section 3.6: ( )p p p cap p (E) = inf{∥φ∥ B 1/ : φ ∈ C(T) ∩ B 1/ . p p , |φ| ≥ 1} For p = 2 it is just the usual logarithmic capacity. Theorem 3.2. Let 1 < p < +∞ and E be a closed subset of T with cap p (E) = 0. Then 1/ p there exists a function f in P+ B p such that partial sums of its Taylor series approximate any continuous function on E. 1/ p

Proof. The shift operator f −→ z· f is bounded on P+ B p . The result follows by Theorem 2.1 1/ p if we prove that the pair (E, P+ ( B p ) satisfies )∗ the S A-property. To check this we take a pair 1/ p (µ, Φ) with µ ∈ M(T) and Φ ∈ P+ B p which annihilates the pairs (z n , z n ), n = 0, 1, . . .. −1/ p

−1/ p

Then P+ µ ∈ B p′ implying by Theorem 3.1 that µ ∈ B p′ . In other words µ has finite p-energy. Since cap p (E) = 0 this implies that µ = 0 and, finally, Φ = 0. □ References [1] H.-P. Beise, J. Müller, Generic boundary behaviour of Taylor series in Hardy and Bergman spaces, Math. Z. 284 (3–4) (2016) 1185–1197. [2] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1) (1966) 135–157. [3] J.B. Garnett, Bounded Analytic Functions, Springer, 2007. [4] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood cliffs, N.J., 1962. [5] S.V. Hruscev, The problem of simultaneous approximation and removal of singularities of Cauchy type integrals, Tr. Mat. Inst. Steklova 130 (1978) 124–195 (in Russian), English trans. in Proceedings of the Steklov Institute of Mathematics, 1979, p. 4. [6] S.V. Hruscev, V. Peller, Hankel Operators, best approximation, and stationary Gaussian processes, Russian Math. Surv. 67 (1982) 61–144. [7] C. Papachristodoulos, M. Papadimitrakis, On universality and convergence of the fourier series of functions in the disc algebra, J. Anal. Math. 137 (1) (2019) 57–71.