On the absolute divergence of Fourier series in the infinite dimensional torus

TL;DR

This paper demonstrates that in infinite-dimensional tori, smooth functions can have Fourier series that diverge absolutely, contrasting with the finite-dimensional case where smoothness implies absolute convergence.

Contribution

It provides simple counterexamples showing that smoothness does not guarantee absolute convergence of Fourier series in infinite dimensions.

Findings

Counterexamples of divergence for smooth functions

Difference between finite and infinite-dimensional Fourier series behavior

Establishment of divergence despite smoothness in infinite dimensions

Abstract

We present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication $f\in C^{(\infty}(\mathbb{T}^\omega)\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^\infty}|\widehat{f}(\bar{p})|<\infty$ is false. There are functions of the class $C^{(\infty}(\mathbb{T}^\omega)$ (depending on an infinite number of variables) whose Fourier series diverges absolutely. This fact establishes a significant difference from the finite dimensional case.

Equations

$$\sum_{m\in\mathbb{Z}^{n}}\big{|}\widehat{f}(m)\big{|}<\infty.$$

$$\sum_{m\in\mathbb{Z}^{n}}(1+|m|)^{N}\big{|}\widehat{f}(m)\big{|}<\infty\quad\forall N=0,1,\ldots,\quad|m|=\Bigl{(}\sum_{i=1}^{n}m_{i}^{2}\Bigr{)}^{1/2}.$$

$$\mathcal{D}(\mathbb{T}^{\omega})=\bigcup_{n=1}^{\infty}\bigl{\{}g_{n}\circ\pi_{n}|g_{n}\in C^{(\infty}(\mathbb{T}^{n})\bigr{\}}$$

$$\hat{f}(\bar{n})={\int }_{{\mathbb{T}}^{\omega }}f(x)e^{-2\pi i\bar{n}\cdot x}dx(\bar{n}\in {\mathbb{Z}}^{\infty })$$

$$\sum _{\bar{n}\in {\mathbb{Z}}^{\infty }}\hat{f}(\bar{n})e^{2\pi i\bar{n}\cdot x}.$$

$$\sum_{\bar{p}\in\mathbb{Z}^{\infty}}(1+|\bar{p}|)^{N}\big{|}\widehat{\phi}(\bar{p})\big{|}<\infty\quad\forall N=0,1,\ldots,\quad|\bar{p}|=\Bigl{(}\sum_{i=1}^{\infty}p_{i}^{2}\Bigr{)}^{1/2}.$$

$$\left|x_j-x_j^0\right|<\delta (j=1,2,\dots ,m),$$

$$\left|f(x_1,x_2,\dots )-f(x_1^0,x_2^0,\dots )\right|<\epsilon .$$

$$C^{(0}({\mathbb{T}}^{\omega })=\left\{f:{\mathbb{T}}^{\omega }\to \mathbb{C}|f\text{ is continuous at all }x\in {\mathbb{T}}^{\omega }\right\}$$

$$\Psi (x)=\sum _{j=1}^{\infty }{a}_j\phi (x_j)$$

$$\sum _{j=m_1+1}^N\left|a_j\right|<\frac{\epsilon }{4{\Vert \phi \Vert }_{\infty }}\text{for all }N>m_1.$$

$$\bigl{|}\varphi(x_{j})-\varphi(x_{j}^{(0)})\bigr{|}<\frac{\varepsilon}{2m_{1}\left\lvert a_{j}\right\rvert}.$$

$$\displaystyle\biggl{|}\sum_{j=1}^{N}a_{j}\varphi(x_{j})$$

$$\displaystyle\leq\sum_{j=1}^{m_{1}}\left\lvert a_{j}\right\rvert\bigl{|}\varphi(x_{j})-\varphi(x_{j}^{(0)})\bigr{|}+\sum_{j=m_{1}+1}^{N}\left\lvert a_{j}\right\rvert\bigl{|}\varphi(x_{j})-\varphi(x_{j}^{(0)})\bigr{|}$$

$$\le \sum _{j=1}^{m_1}\left|a_j\right|\cdot \frac{\epsilon }{2m_1\left|a_j\right|}+2{\Vert \phi \Vert }_{\infty }\sum _{j=m_1+1}^N\left|a_j\right|<\frac{\epsilon }{2}+\frac{\epsilon }{2}=\epsilon .$$

$$\bigl{|}\Psi(x)-\sum_{j=1}^{N}a_{j}\varphi(x_{j})\bigr{|}<\varepsilon\quad\forall x\in\mathbb{T}^{\omega}.$$

$$\displaystyle\bigl{|}\Psi(x)-\Psi(x^{(0)})\bigr{|}$$

$$\displaystyle\qquad+\bigl{|}\Psi(x^{(0)})-\sum_{j=1}^{M}a_{j}\varphi(x_{j}^{(0)})\bigr{|}$$

$$<3\epsilon$$

$$D^{\alpha }f=D_{j_1}^{{\alpha }_{j_1}}\cdots D_{j_m}^{{\alpha }_{j_m}}f=\frac{{\partial }^{{\alpha }_{j_1}}}{\partial x_{j_1}^{{\alpha }_{j_1}}}\cdots \frac{{\partial }^{{\alpha }_{j_m}}}{\partial x_{j_m}^{{\alpha }_{j_m}}}f\text{if αj=0 for j∈/{j1,…,jm}.}$$

$${\Vert f\Vert }_{(k)}=\underset{0\le |\alpha |\le k}{\sup }{\Vert D^{\alpha }f\Vert }_{\infty }$$

$$\sum _{m,n=1}^{\infty }{a}_{mn}.$$

$$s_{MN}:=\sum _{m=1}^M\sum _{n=1}^N{a}_{mn},(M,N)\in {\mathbb{N}}^2.$$

$$|s_{MN}-s|<\epsilon \text{if M,N≥μ}.$$

$$\forall \epsilon >0\exists \mu :|s_{PQ}-s_{MN}|<\epsilon \text{ if }P>M\ge \mu \text{ and }Q>N\ge \mu .$$

$$\sum _{m,n}(a_{mn}+b_{mn})=\sum _{m,n}{a}_{mn}+\sum _{m,n}{b}_{mn}.$$

$$\sum _{m,n=1}^{\infty }{a}_{mn}=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{a}_{mn}=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{a}_{mn}$$

$$z=e^{2\pi ix}:=\bigl{(}e^{2\pi ix_{1}},\ldots,e^{2\pi ix_{n}},\ldots\bigr{)}\in\mathcal{S}.$$

$$Q(x,y):=\sum _{m,n=1}^{\infty }{a}_{mn}{x}_m{y}_n(a_{mn}\in \mathbb{C},x,y\in \mathcal{S}).$$

$$\biggl{|}\sum_{m=1}^{M}\sum_{n=1}^{N}a_{mn}x_{m}y_{n}\biggr{|}\leq H\quad\forall x,y\in\mathcal{S},\ \forall M,N\in\mathbb{N}.$$

$$Q_{MN}(x,y):=\sum _{m=1}^M\sum _{n=1}^N{a}_{mn}{x}_m{y}_n(x,y\in \mathcal{S}).$$

$$x^{(M)}:=(x_1,\dots ,x_M),y^{(N)}:=(y_1,\dots ,y_N).$$

Full text

1. Introduction

The following result establishes a sufficient condition of smoothness on a function defined on the $n$-dimensional torus $\mathbb{T}^{n}$ ($n\geq 1$) for the absolute convergence of its Fourier series:

Theorem 1.1**.**

([13, p. 249].)* If $f\in C^{(k}(\mathbb{T}^{n})$, $k>n/2$, then*

[TABLE]

When $f\in C^{(\infty}(\mathbb{T}^{n})$, more conclusive results hold, for example (see [12, Th. 7.25, p. 202]):

Theorem 1.2**.**

If $f\in C^{(\infty}(\mathbb{T}^{n})$, then

[TABLE]

This same result holds for cylindrical infinitely smooth functions defined on the infinite dimensional torus $\mathbb{T}^{\omega}$, the compact abelian group which is the complete direct sum of countably many copies of $\mathbb{T}\simeq\mathbb{R}/\mathbb{Z}$. Recall that $f$ is a cylindrical function on $\mathbb{T}^{\omega}$ if $f$ depends only on a finite number of variables, i.e., there exists $n\geq 1$ and $g_{n}\colon\Omega_{n}\to\mathbb{C}$, with $\Omega_{n}\subseteq\mathbb{T}^{n}$, such that $f=g_{n}\circ\pi_{n}$, with $\pi_{n}\colon\mathbb{T}^{\omega}\to\mathbb{T}^{n}$ being the canonical projection. The space of cylindrical functions of class $C^{(\infty}$ on $\mathbb{T}^{\omega}$ (see Definition 2.3) is defined ([1, p. 73–75]) by

[TABLE]

so that if $f\in\mathcal{D}(\mathbb{T}^{\omega})$, then there exist $p\in\mathbb{N}$ and $g_{p}\in C^{(\infty}(\mathbb{T}^{p})$ such that $f=g_{p}\circ\pi_{p}$.

The dual group of $\mathbb{T}^{\omega}$, denoted by $\mathbb{Z}^{\infty}$, is the direct sum of countably many copies of $\mathbb{Z}$, formed by the finitely nonzero sequences of integer numbers. Denote by $dx$ the normalized Haar measure on $\mathbb{T}^{\omega}$. If $f\in L^{1}(\mathbb{T}^{\omega})$, then the function $\widehat{f}$ defined on $\mathbb{Z}^{\infty}$ by

[TABLE]

is the Fourier transform of $f$, the Fourier series of $f$ being the formal series (observe that $\mathbb{Z}^{\infty}$ is a countable set)

[TABLE]

By using the ideas in the proof of Theorem 1.2, the following result can be proved (see also [2, Proposition 1]):

Theorem 1.3**.**

If $\phi\in\mathcal{D}(\mathbb{T}^{\omega})$, then

[TABLE]

In May 2016, in a private communication to the second author, Professor A. D. Bendikov conjectured that the implication $f\in C^{(\infty}(\mathbb{T}^{\omega})\Rightarrow\sum_{\bar{p}\in\mathbb{Z}^{\infty}}|\widehat{f}(\bar{p})|<\infty$, which holds, as already mentioned, for functions depending only on a finite number of variables, is in general false. This would mean a significant difference to what happens in the finite-dimensional case.

In this note we confirm Bendikov’s conjecture by producing some counterexamples via quadratic forms depending on an infinite number of variables. The construction of such counterexamples is based on classical results of Toeplitz [14], Littlewood [10] and Bohnenblust and Hille [4]111Prof. Bendikov suggested that a counterexample could be constructed through an appropriate Jacobi theta function in an infinite number of variables. Our construction is different..

The main result in this note is the following.

Theorem 1.4**.**

There exist functions of the class $C^{(\infty}(\mathbb{T}^{\omega})$ (depending on an infinite number of variables) whose Fourier series diverges absolutely.

Although we restrict ourselves to the case of the infinite dimensional torus, we point out that Bendikov and L. Saloff-Coste [3] have studied several scales of smooth functions in the more general setting of connected infinite-dimensional compact groups.

In Section 2 we introduce some definitions and give several basic results. In Section 3 we present a detailed account of bilinear and quadratic forms in an infinite number of variables used to construct our counterexamples. The proof of Theorem 1.4 and the counterexamples are given in Section 4.

2. Premilinary definitions and results

We begin by providing some basic principles.

Definition 2.1**.**

([5, p. 130].) The function $f\colon\mathbb{T}^{\omega}\to\mathbb{C}$ is continuous at the point $x^{(0)}=(x_{1}^{0},x_{2}^{0},\ldots)$ if for every $\varepsilon>0$ there is a positive integer $m$ and a number $\delta>0$ such that for each $(x_{1},x_{2},\ldots)\in\mathbb{T}^{\omega}$ satisfying

[TABLE]

we have

[TABLE]

Since $\mathbb{T}^{\omega}$ is compact, the vector space

[TABLE]

is a Banach space with the norm $\left\lVert f\right\rVert_{\infty}=\max_{x\in\mathbb{T}^{\omega}}\left\lvert f(x)\right\rvert$.

Lemma 2.2**.**

Let $\varphi(t)\in C^{(0}(\mathbb{T})$ and $\sum_{j=1}^{\infty}a_{j}$ be an absolutely convergent series of complex numbers. Then the function

[TABLE]

is continuous on $\mathbb{T}^{\omega}$.

Proof.

We can clearly suppose that $\varphi$ is not the zero function. Fix $x^{(0)}\in\mathbb{T}^{\omega}$. Given $\varepsilon>0$, since $\sum_{j=1}^{\infty}\left\lvert a_{j}\right\rvert$ converges, there exists $m_{1}\in\mathbb{N}$ such that

[TABLE]

On the other hand, for each $j=1,\ldots,m_{1}$, the continuity of $\varphi$ at $x_{j}^{(0)}$ ensures the existence of $\delta_{j}>0$ such that if \bigl{|}x_{j}-x_{j}^{(0)}\bigr{|}<\delta_{j}, then

[TABLE]

Let $\delta=\min_{1\leq j\leq m_{1}}\delta_{j}$. If $x\in\mathbb{T}^{\omega}$ satisfies \bigl{|}x_{j}-x_{j}^{(0)}\bigr{|}<\delta for $j=1,\ldots,m_{1}$, then for all $N>m_{1}$,

[TABLE]

Moreover, $\sum_{j=1}^{\infty}\left\lvert a_{j}\varphi(x_{j})\right\rvert\leq\left\lVert\varphi\right\rVert_{\infty}\sum_{j=1}^{\infty}\left\lvert a_{j}\right\rvert$ for all $x\in\mathbb{T}^{\omega}$. Therefore, the series defining $\Psi(x)$ is absolutely convergent, thus $\Psi(x)$ is defined for all $x\in\mathbb{T}^{\omega}$ and there exists $m_{2}=m_{2}(\varepsilon)$ such that, if $N>m_{2}$, then

[TABLE]

Consequently, taking $M=\max\{m_{1},m_{2}\}$, we have

[TABLE]

if $x\in\mathbb{T}^{\omega}$ satisfies \bigl{|}x_{j}-x_{j}^{(0)}\bigr{|}<\delta for $j=1,\ldots,M$, and therefore $\Psi(x)$ is continuous at $x^{(0)}$. ∎

Definition 2.3**.**

Let $f$ be a function defined on $\mathbb{T}^{\omega}$. For each multiindex $\alpha=(\alpha_{1},\alpha_{2},\ldots)$ which is finitely nonzero, that is, $\alpha_{j}\neq 0$ for only finitely many $j$, the partial differentiation operator $D^{\alpha}$ is defined by

[TABLE]

The total order of $\alpha$ is $\left\lvert\alpha\right\rvert=\alpha_{j_{1}}+\ldots+\alpha_{j_{m}}$. When $\left\lvert\alpha\right\rvert=0$, $D^{\alpha}f=f$.

For each $k$, $C^{(k}(\mathbb{T}^{\omega})$ is defined as the class of all functions $f$ with continuous partial derivatives up to the $k$-th order, i.e., $D^{\alpha}f\in C^{(0}(\mathbb{T}^{\omega})$ for all finitely nonzero multiindices $\alpha$ such that $\left\lvert\alpha\right\rvert\leq k$. With the norm

[TABLE]

where $\left\lVert D^{\alpha}f\right\rVert_{\infty}=\max_{x\in\mathbb{T}^{\omega}}\left\lvert(D^{\alpha}f)(x)\right\rvert$ for each fixed $\alpha$, $C^{(k}(\mathbb{T}^{\omega})$ is a Banach space [7, 2.2.4]. The space of infinitely differentiable functions is the intersection $C^{(\infty}(\mathbb{T}^{\omega})=\bigcap_{k=0}^{\infty}C^{(k}(\mathbb{T}^{\omega})$ and it is a Fréchet space [7, 12.1].

Double series. (See [6, pp. 72-76], and also [11].) Consider a double series of complex numbers,

[TABLE]

Rectangular partial (finite) sums of (2.1) are

[TABLE]

The series (2.1) is said to converge to $s\in\mathbb{C}$ in Pringsheim’s sense when for every $\varepsilon>0$ there exists $\mu$ such that

[TABLE]

A necessary and sufficient condition for the convergence of (2.1) in Pringsheim’s sense is:

[TABLE]

When the series $\sum_{m,n}a_{mn}$ and $\sum_{m,n}b_{mn}$ converge in Pringsheim’s sense, so does $\sum_{m,n}(a_{mn}+b_{mn})$, and

[TABLE]

Hardy [8, p. 88] introduced the notion of regular convergence of a double series as follows: the series (2.1) is said to converge regularly to $s\in\mathbb{C}$ if it converges to $s$ in Pringsheim’s sense and, in addition, each of its row and column series, $\sum_{n=1}^{\infty}a_{mn}$ for each $m=1,2,\ldots$, and $\sum_{m=1}^{\infty}a_{mn}$ for each $n=1,2,\ldots$, also converges as a single series.

An absolutely convergent double series is also regularly convergent, and regular convergence is sufficient for

[TABLE]

hold [11, Th. 1].

3. Bilinear and quadratic forms in an infinite number of variables

Let us denote by \mathcal{S}:=\bigl{\{}(z_{n})_{n=1}^{\infty}\,|\,z_{n}\in\mathbb{C},\ \left\lvert z_{n}\right\rvert\leq 1\ \forall n\in\mathbb{N}\bigr{\}} the infinite-dimensional polydisc (the closed unit ball of $\ell_{\infty}(\mathbb{N})$). Analogously to $\mathbb{T}^{\omega}$, we will consider the space $\mathcal{S}$ with the topology of the cartesian product of infinitely many closed unit circles of the complex plane. In particular, if $x\in\mathbb{T}^{\omega}$, then

[TABLE]

We define a bilinear form in $\mathcal{S}$ (in principle only formally) by the expression

[TABLE]

The bilinear character and the very existence of the function $Q(x,y)$ depend on the convergence of the double series above.

Definition 3.1**.**

The series (3.1) is completely bounded in $\mathcal{S}$ if there is a constant $H$ such that

[TABLE]

The following property is immediate.

Lemma 3.2**.**

Suppose that the series (3.1) is completely bounded in $\mathcal{S}$. Then the series $\sum_{n=1}^{\infty}\left\lvert a_{mn}\right\rvert$ for each $m\in\mathbb{N}$, and $\sum_{m=1}^{\infty}\left\lvert a_{mn}\right\rvert$ for each $n\in\mathbb{N}$, are convergent.

Proof.

For $M,N\in\mathbb{N}$, let $Q_{MN}(x,y)$ denote the rectangular partial sum or section

[TABLE]

The section $Q_{MN}(x,y)$ only depends on the first $M$ components of $x$ and on the first $N$ components of $y$, and thus we can consider it as a bilinear form on $D^{M}\times D^{N}$, where $D$ denotes the closed unit disc of the complex plane. Let us write

[TABLE]

Then by hypothesis we have

[TABLE]

Fix $n_{0}\in\mathbb{N}$ (when we fix $m_{0}\in\mathbb{N}$, we proceed in a similar way). Consider the points

[TABLE]

($\delta_{ij}$ is the Kronecker’s symbol). Obviously, $\xi_{n_{0}}$ and $\eta_{n_{0}}$ belongs to $\mathcal{S}$, and for each $M\in\mathbb{N}$ such that $M>n_{0}$,

[TABLE]

with $H$ independent of $M$. Consequently the series $\sum_{m=1}^{\infty}\left\lvert a_{mn_{0}}\right\rvert$ is convergent. ∎

The theorem which follows is due to Littlewood [10, p. 166-168].

Theorem 3.3**.**

If the series (3.1) is completely bounded in $\mathcal{S}$ by a constant $H$, then it converges in Pringsheim’s sense, uniformly in $\mathcal{S}^{2}$, to a bilinear form $Q(x,y)$ which satisfies $|Q(x,y)|\leq H$ for all $x,y\in\mathcal{S}$ (we then say that the bilinear form $Q(x,y)$ is completely bounded on $\mathcal{S}$).

Observe that $Q(x,y)$ is a bilinear form if for all $x,x^{\prime},y,y^{\prime}\in\mathcal{S}$ it is verified

[TABLE]

If the bilinear form $Q(x,y)$ is completely bounded on $\mathcal{S}$, then in particular, given $\varepsilon>0$, there exists $\nu_{1}=\nu_{1}(\varepsilon)$ such that \bigl{|}Q(x,y)-Q_{\nu\nu}(x,y)\bigr{|}<\varepsilon for all $(x,y)\in\mathcal{S}^{2}$ and $\nu\geq\nu_{1}$. From this we easily deduce:

Corollary 3.4**.**

A bilinear form $Q(x,y)$ completely bounded on $\mathcal{S}$ defines a continuous function on $\mathcal{S}^{2}$.

Proof.

Fix $(x_{0},y_{0})\in\mathcal{S}^{2}$ and $\varepsilon>0$. First, as just said above, there exists $\nu_{1}(\varepsilon)$ such that

[TABLE]

if $\nu\geq\nu_{1}$. On the other hand, the bilinear form defined on $D^{\nu_{1}}\times D^{\nu_{1}}$ by

[TABLE]

is continuous at \bigl{(}x_{0}^{(\nu_{1})},y_{0}^{(\nu_{1})}\bigr{)}. Thus, there exists $\delta>0$ (depending on $(x_{0},y_{0})$ and $\varepsilon$) such that, if \max_{1\leq j\leq\nu_{1}}\bigl{\{}|x_{j}-x_{0j}|,|y_{j}-y_{0j}|\bigr{\}}<\delta, then

[TABLE]

Consequently, for every $(x,y)\in\mathcal{S}^{2}$ with \max\bigl{\{}|x_{j}-x_{0j}|,|y_{j}-y_{0j}|\bigr{\}}<\delta for $j=1,\ldots,\nu_{1}$, we have

[TABLE]

(we have used that Q_{\nu_{1}\nu_{1}}(x,y)=Q_{\nu_{1}\nu_{1}}\bigl{(}x^{(\nu_{1})},y^{(\nu_{1})}\bigr{)} for all $(x,y)\in\mathcal{S}^{2}$), and the continuity of $Q(x,y)$ at $(x_{0},y_{0})$ is proved. ∎

Some more definitions and remarks. Let $Q(x,y)=\sum_{m,n=1}^{\infty}a_{mn}x_{m}y_{n}$ be a bilinear form completely bounded by a constant $H$ in $\mathcal{S}$, and which defines, according to Corollary 3.4, a continuous function on $\mathcal{S}^{2}$. Let

[TABLE]

for $x\in\mathcal{S}$. The quadratic form $C(x)$ is also called completely bounded in $\mathcal{S}$, because

[TABLE]

When the bilinear form $Q(x,y)$ is completely bounded in $\mathcal{S}$, its partial derivatives are well defined (see [9, p. 128]). Writing, for each $p\in\mathbb{N}$, $e_{p}=(\delta_{pn})_{n=1}^{\infty}\in\mathcal{S}$, and applying (3.3), we have

[TABLE]

and thus these partial derivatives are bounded linear forms. According to Lemma 3.2, the series $\sum_{n=1}^{\infty}\left\lvert a_{pn}\right\rvert$ and $\sum_{m=1}^{\infty}\left\lvert a_{mp}\right\rvert$ are convergent for all $p$, and hence one can deduce that $\frac{\partial Q}{\partial x_{p}}(x,y)$ and $\frac{\partial Q}{\partial y_{p}}(x,y)$ are continuous on $\mathcal{S}^{2}$ by applying a result (on $\mathcal{S}^{2}$) analogous to Lemma 2.2 (in $\mathbb{T}^{\omega}$).

Corollary 3.5**.**

(a)* If the bilinear form $Q(x,y)$ is completely bounded on $\mathcal{S}$, then the quadratic form $C(x)=Q(x,x)$ belongs to the class $C^{(\infty}(\mathcal{S})$.*

(b)* If the quadratic form $C(x)=Q(x,x)$ is completely bounded in $\mathcal{S}$, then it belongs to the class $C^{(\infty}(\mathcal{S})$.*

Proof.

(a) The quadratic form $C(x)=Q(x,x)$ is continuous in $\mathcal{S}$ according to Corollary 3.4. For each $p\in\mathbb{N}$ we have

[TABLE]

due to the absolute convergence of each series. Then, by applying Lemma 2.2, the linear form $\frac{\partial C}{\partial x_{p}}(x)$ is continuous on $\mathcal{S}$, and its partial derivatives are constant functions.

(b) From the identity

[TABLE]

it follows that the bilinear form $Q(x,y)$ is completely bounded on $\mathcal{S}$. Then apply part (a). ∎

4. Functions in $C^{(\infty}(\mathbb{T}^{\omega})$ whose Fourier series diverges absolutely

In this section, we prove Theorem 1.4. In 1913, Toeplitz [14, p. 427] introduced a quadratic form

[TABLE]

in infinitely many variables, symmetric (i.e., $a_{mn}=a_{nm}$), completely bounded on $\mathcal{S}$ in the above sense, and such that the series $\sum_{m,n=1}^{\infty}|a_{mn}|$ diverges. This quadratic form will be described below. We will simply replace $z=e^{2\pi ix}$ (i.e., $z_{j}=e^{2\pi ix_{j}}$ for all $j$) with $x\in\mathbb{T}^{\omega}$ in Toeplitz’s form, and consider the function

[TABLE]

From Corollary 3.5 (b) it follows that $F\in C^{(\infty}(\mathbb{T}^{\omega})$. In particular, $F$ is integrable.

Let us now calculate the Fourier coefficients of $F$. We will use $F(x)=\lim_{M\to\infty}F_{M}(x)$, where

[TABLE]

Since the quadratic form (4.1) is completely bounded on $\mathcal{S}$, i.e., $|C(z)|\leq H$ for all $z\in\mathcal{S}$, we have $\left\lvert F_{M}(x)\right\rvert\leq H$ for all $M\in\mathbb{N}$ and $x\in\mathbb{T}^{\omega}$. This allows to apply Vitali’s convergence theorem to write, for any $\bar{p}\in\mathbb{Z}^{\infty}$ fixed,

[TABLE]

[TABLE]

where we denote by $\bar{e}_{q}$ the element $(\delta_{qj})_{j=1}^{\infty}$ belonging to $\mathbb{Z}^{\infty}$.

Thus, the expression (4.2), which defines $F(x)$, is indeed its Fourier series, $\sum_{\bar{p}\in\mathbb{Z}^{\infty}}\widehat{F}(\bar{p})e^{2\pi i\bar{p}\cdot x}$. Therefore

[TABLE]

and since $\sum_{m,n=1}^{\infty}\left\lvert a_{mn}\right\rvert=\infty$, our function $F$ is a counterexample showing that the implication $f\in C^{(\infty}(\mathbb{T}^{\omega})\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^{\infty}}|\widehat{f}(\bar{p})|<\infty$ is false.

Let us proceed to describe the quadratic form $C(z)$. We first show an auxiliary lemma. Toeplitz [14, p. 423-426] gave it for real orthogonal matrices. In what follows, $D$ denotes the closed unit disc of the complex plane.

Lemma 4.1**.**

(Littlewood, [10, p. 171]; see also [4, p. 609].) Let $A=(a_{mn})_{N\times N}$ be a unitary matrix, i.e.,

[TABLE]

and define $Q_{NN}(x):=N^{-1}\sum_{m,n=1}^{N}a_{mn}x_{m}x_{n}$ for $x\in D^{N}$. Then,

[TABLE]

Toeplitz’s quadratic form. Toeplitz begins by defining $C_{1}(z_{1},\ldots,z_{4})$ as the quadratic form in $D^{4}$ whose coefficient matrix is

[TABLE]

The real symmetric matrix $C_{1}$ verifies $C_{1}^{2}=4I$, and so, by Lemma  4.1,

[TABLE]

in $D^{4}$ (this maximum value is attained for $z_{1}=\ldots=z_{4}=1$).

Next, he defines $C_{2}(z_{1},\ldots,z_{4^{2}})$ as the quadratic form in $D^{4^{2}}$ whose coefficient matrix is

[TABLE]

Lemma 4.1 yields

[TABLE]

in $D^{4^{2}}$ (and the maximum modulus is attained for $z_{1}=\ldots=z_{4^{2}}=1$).

Inductively, from the quadratic form in $4^{\alpha}$ variables ($\alpha\geq 1$) with matrix $C_{\alpha}$, one can construct the quadratic form in $4^{\alpha+1}$ variables with matrix

[TABLE]

According to Lemma 4.1 we have that, for all $\alpha\in\mathbb{N}$,

[TABLE]

in $D^{4^{\alpha}}$. Finally, for $x\in\mathcal{S}$, Toeplitz defines

[TABLE]

where $(\mu_{\alpha})_{\alpha=1}^{\infty}$ is a sequence of positive numbers determined below, and he shows the following lemma (see [14, p. 426-427]):

Lemma 4.2**.**

If $\mu_{\alpha}>0$ are chosen so that the series $\sum\mu_{\alpha}$ is convergent, then the quadratic form (4.3) is completely bounded on $\mathcal{S}$.

Moreover, the sum of the moduli of all coefficients of the form $C(x)$ is $\sum 2^{\alpha}\mu_{\alpha}$. It is easy to choose $\mu_{\alpha}$ so that $\sum\mu_{\alpha}<\infty$ and $\sum 2^{\alpha}\mu_{\alpha}=\infty$ (for example, $\mu_{\alpha}=\frac{1}{\alpha^{2}}$, $\mu_{\alpha}=2^{-\alpha}$, etc.). Thus, the function

[TABLE]

constructed with these $\mu_{\alpha}$ is our first announced counterexample.

Littlewood’s quadratic forms. From [10, p. 171-173] and [4, p. 609-612] we can get a variety of counterexamples that generalize the preceding one, based on quadratic forms on $\mathcal{S}$ for which not all the coefficients are real.

For example, let $N>2$ be a fixed integer, and consider the infinite collection of matrices

[TABLE]

All entries in $M_{\mu}$ are $N$th roots of unity, and $M_{\mu}$ is an unitary matrix for all $\mu\in\mathbb{N}$. Let us denote by M_{\mu}\bigl{(}x_{1}^{(\mu)},\ldots x_{N^{\mu}}^{(\mu)}\bigr{)} the quadratic form associated with the matrix $M_{\mu}$ and the variables of a generic point $x\in\mathcal{S}$ on which it acts, and then define the quadratic form in infinitely many variables

[TABLE]

According to Lemma 4.1 we have

[TABLE]

so that

[TABLE]

Thus, $M(x)$ is completely bounded and, by Corollary 3.5 (b), it belongs to the class $C^{(\infty}$ on $\mathcal{S}$. But, if we denote $M(x)=\sum_{m,n=1}^{\infty}a_{mn}x_{m}x_{n}$, since all the moduli of the nonzero coefficients are equal to 1, we have

[TABLE]

and so the Fourier series of the function $G(x)=M(e^{2\pi ix})$, $x\in\mathbb{T}^{\omega}$, diverges absolutely.

Bohnenblust and Hille [4, p. 608-614] generalized the results of Littlewood to $m$-variate forms ($m>2$). This would provide new counterexamples, this time based on $m$-variate forms ($m>2$) in infinitely many variables.

Acknowledgments

The authors are grateful to Professor A. D. Bendikov for the suggested ideas and the careful reading of the manuscript.

Luz Roncal was supported by the Basque Government through the BERC 2018–2021 program, by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323, through project MTM2015-65888-C04-4-P, and through project MTM2017-82160-C2-1-P funded by (AEI/FEDER, UE) and acronym

“HAQMEC”, and by a 2017 Leonardo grant for Researchers and Cultural Creators, BBVA Foundation. The Foundation accepts no responsibility for the opinions, statements and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors.