N. A. Dyuzhina, “Density of the sums of shifts of a single function in the $L_2^0$ space on a compact Abelian group”, Sb. Math., 215:6 (2024), 743

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Sbornik: Mathematics, 2024, Volume 215, Issue 6, Pages 743–754
DOI: https://doi.org/10.4213/sm10011e (Mi sm10011)

This article is cited in 1 scientific paper (total in 1 paper)

Density of the sums of shifts of a single function in the

L_{2}^{0}

$L_2^0$ space on a compact Abelian group

N. A. Dyuzhina^ab

^a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

English version PDF (471 kB) HTML full-text Citations (1) Russian version article

References:

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DOI: https://doi.org/10.4213/sm10011e

Abstract: Let

G

$G$ be a nontrivial compact Abelian group. The following result is proved: a real-valued function on

G

$G$ such that the sums of shifts of it are dense in the

L_{2}

$L_{2}$ -norm in the corresponding real space of mean zero functions exists if and only if the group

G

$G$ is connected and has an infinite countable character group.
Bibliography: 13 titles.

Keywords: density, sums of shifts, compact groups, space

L_{2}

$L_{2}$ .

Funding agency	Grant number
Foundation for the Advancement of Theoretical Physics and Mathematics BASIS	20-8-3-5-1
This research was carried out with the support of the Theoretical Physics and Mathematics Advancement Foundation “BASIS” (grant no. 20-8-3-5-1).

Received: 08.10.2023

Bibliographic databases:

Document Type: Article

MSC: 41A46, 43A15

Language: English

Original paper language: Russian

§ 1. Introduction

In the real space $L^{0}_{p}(\mathbb{T})$ of mean zero functions on the circle $\mathbb{T}$ that are integrable to power $p$ , $1\leqslant p <\infty$ , there exists a function $f$ such that the sums of shifts of $f$ are dense in $L^{0}_{p}(\mathbb{T})$ (see [1], where whole classes of such functions were distinguished). There exists a function on the real line $\mathbb{R}$ such that the sums of shifts of it are dense in all real spaces $L_{p}(\mathbb{R})$ for $2 \leqslant p <\infty$ [2]. The real space $l_{2}(\mathbb{Z})$ of two-sided sequences contains an element such that the sums of its shifts are dense in all real spaces $l_{p}(\mathbb{Z})$ for $2 \leqslant p <\infty$ (see [3]). In [4] these results were extended to several dimensions, that is, to the torus $\mathbb{T}^{d}$ , the space $\mathbb{R}^{d}$ and the lattice $\mathbb{Z}^{d}$ , where $d \in \mathbb{N}$ . In [5] sufficient conditions on a function $f$ were found under which the sums of functions of the form $f(\alpha x - \theta)$ , $\alpha \in A \subset \mathbb{R}$ , $\theta \in \Theta \subset \mathbb{R}$ , are dense in the real space $L_{p}$ on the line or a compact subset of it. For a survey of results on the density of the sums of shifts of a single function, see [6]. In this connection the following problem arises in the natural way, which was stated in [6].

Problem 1. Let $G$ be a locally compact Abelian group with Haar measure $m$ . Does there exist a function $f$ on this group such that the sums

\begin{matrix} (1.1) & \sum_{k = 1}^{n} f (g + g_{k}), g_{k} \in G, n \in N, \end{matrix}

$\begin{equation} \sum_{k=1}^{n} f(g+g_{k}), \qquad g_{k} \in G, \quad n \in \mathbb{N}, \end{equation} \tag{1.1}$

of shifts of

f

$f$ :

(a) are dense in the real space $L_{2}(G)$ (when $G$ is noncompact), or
(b) are dense in the real space
$\begin{equation*} L_{2}^{0}(G)=\biggl\{h \in L_{2}(G)\colon \int_{G}h(g)\,dm(g)=0 \biggr\} \end{equation*} \notag$
(when $G$ is compact)?

The compact case is distinguished in Problem 1 for the following reason. If $G$ is a compact Abelian group, then its Haar measure satisfies $m(G) < \infty$ (see [7], Ch. 4, § 15, Theorem 15.9). Then for a function $f \in L_{2}(G)$ the sums (1.1) of shifts of $f$ cannot be dense in the whole of $L_{2}(G)$ : the mean value $\displaystyle \int_{G}f(g)\,dm(g)=\alpha$ of $f$ is defined and sums (1.1) cannot approximate functions with mean value outside the set $\{n\alpha\colon n \in \mathbb{N} \}$ .

The aim of this paper is to give an exhaustive answer to question (b) in Problem 1, namely, establish the following result.

Theorem 1. Let $G$ be a nontrivial compact Abelian group. Then a function $f$ : $G \to \mathbb{R}$ such that the sums (1.1) of shifts of $f$ are dense in the real space $L_{2}^{0}(G)$ exists if and only if $G$ is connected and its character group is infinite countable.

Note that if $G$ is a trivial group consisting of the single neutral element, then the space $L_{2}^{0}(G)$ contains only the function identically equal to zero, the sums of shifts of which are dense in this space.

§ 2. Auxiliary lemmas

Throughout what follows $f_{g}(\,\cdot\,)=f(\,\cdot+g)$ is the shift of $f$ by the element $g$ of $G$ . We prove a few auxiliary lemmas.

Lemma 1. If $G$ is a finite nontrivial Abelian group, then there exists no function $f\colon G \to \mathbb{R}$ such that the sums (1.1) of shifts of it are dense in the real space $L_{2}^{0}(G)$ .

Proof. We fix a real function

f \in L_{2}^{0} (G)

$f \in L_{2}^{0}(G)$ and number the elements of the group:

G = {g_{1}, g_{2}, \dots, g_{N}}

$G= \{g_{1}, g_{2}, \dots , g_{N} \}$ ,

N ⩾ 2

$N \geqslant 2$ . Each

g_{k}

$g_{k}$ is an atom of the measure

m

$m$ , and

m (g_{k}) = 1 / N

$m(g_{k})=1/N$ . We represent a function

h : G \to R

$h\colon G \to \mathbb{R}$ as the vector of values

(h (g_{1}), h (g_{2}), \dots, h (g_{N}))

$(h(g_{1}), h(g_{2}), \dots , h(g_{N}))$ , and we represent

L_{2}^{0} (G)

$L_{2}^{0}(G)$ as the space

L

$L$ of vectors of length

N

$N$ with sum of coordinates equal to zero. Since

f

$f$ has mean value zero, for each

g \in G

$g \in G$ we have

\sum_{n = 1}^{N} f (g + g_{n}) = 0

$\sum_{n=1}^{N} f(g+g_{n})=0$ . Therefore,

\begin{matrix} (2.1) & \sum_{n = 1}^{N} f_{g_{n}} = 0. \end{matrix}

$\begin{equation} \sum_{n=1}^{N} f_{g_{n}}=0. \end{equation} \tag{2.1}$

Consider the set of sums of shifts of $f$ :

\begin{matrix} (2.2) & S := {\sum_{k = 1}^{m} f_{h_{k}} : h_{k} \in G, m \in N} = {\sum_{n = 1}^{N} ν_{n} f_{g_{n}} : ν_{n} \in N \cup {0}} . \end{matrix}

$\begin{equation} S :=\biggl\{\sum_{k=1}^{m} f_{h_{k}}\colon h_{k} \in G, \ m \in \mathbb{N} \biggr\} =\biggl\{\sum_{n=1}^{N} \nu_{n}f_{g_{n}}\colon \nu_{n} \in \mathbb{N} \cup \{0\} \biggr\}. \end{equation} \tag{2.2}$

From (2.1) and (2.2) we obtain

\begin{aligned} S & = {\sum_{n = 1}^{N - 1} ν_{n} f_{g_{n}} - ν_{N} (\sum_{n = 1}^{N - 1} f_{g_{n}}) : ν_{n} \in N \cup {0}} \\ = {\sum_{n = 1}^{N - 1} (ν_{n} - ν_{N}) f_{g_{n}} : ν_{n} \in N \cup {0}} = {\sum_{n = 1}^{N - 1} λ_{n} f_{g_{n}} : λ_{n} \in Z} . \end{aligned}

$\begin{equation*} \begin{aligned} \, S&=\biggl\{\sum_{n=1}^{N-1} \nu_{n}f_{g_{n}} - \nu_{N}\biggl( \sum_{n=1}^{N-1} f_{g_{n}} \biggr)\colon \nu_{n} \in \mathbb{N} \cup \{0\} \biggr\} \\ &=\biggl\{\sum_{n=1}^{N-1} (\nu_{n} - \nu_{N})f_{g_{n}}\colon \nu_{n} \in \mathbb{N} \cup \{0\} \biggr\} =\biggl\{\sum_{n=1}^{N-1} \lambda_{n}f_{g_{n}}\colon \lambda_{n} \in \mathbb{Z} \biggr\}. \end{aligned} \end{equation*} \notag$

If the vectors

f_{g_{n}}

$f_{g_{n}}$ ,

n = 1, \dots, N - 1

$n=1,\dots ,N-1$ , are linearly independent, then the set

S

$S$ , and therefore also its closure

\bar{S}

$\overline{S}$ , is the integer lattice generated by these vectors, so the closure

\bar{S}

$\overline{S}$ cannot coincide with the

(N - 1)

$(N-1)$ -dimensional space

L

$L$ . If the vectors

f_{g_{n}}

$f_{g_{n}}$ ,

n = 1, \dots, N - 1

$n=1,\dots ,N-1$ , are linearly dependent, then

\bar{S}

$\overline{S}$ lies in a subspace of dimension at most

N - 2

$N-2$ , so it cannot coincide with

L

$L$ of dimension

N - 1

$N-1$ .

Lemma 1 is proved.

Lemma 2. Let $G$ be a compact Abelian group and $H$ be a closed subgroup of $G$ . If there exists a function $f_{0}\colon G \to \mathbb{R}$ the sums of shifts of which are dense in the real space $L_{2}^{0}(G)$ , then there exists a function $F_{0}\colon G/H \to \mathbb{R}$ the sums of shifts of which are dense in the real space $L_{2}^{0}(G/H)$ .

Proof. Since

G

$G$ is a compact Abelian group and

H

$H$ is a closed subgroup of it,

G / H

$G/H$ is a compact Abelian group (see [7], Ch. 2, § 5, Theorem 5.22, and [8], Appendix B, § B6). We denote the coset of an element

x \in G

$x \in G$ by the subgroup

H

$H$ by

\hat{x}

$\widehat{x}$ . According to [8], Ch. 2, § 2.7.3, the groups

G, H

$G, H$ and

G / H

$G/H$ are endowed with the Haar measures

m_{G}

$m_{G}$ ,

m_{H}

$m_{H}$ and

m_{G / H}

$m_{G/H}$ such that

m_{H} (H) = 1

$m_{H}(H)=1$ and for each function

f \in L_{1} (G)

$f \in L_{1}(G)$ the function

\begin{matrix} (2.3) & F (\hat{x}) = \int_{H} f (x + y) d m_{H} (y) \end{matrix}

$\begin{equation} F(\widehat{x})=\int_{H}f(x+y)\,dm_{H}(y) \end{equation} \tag{2.3}$

is well defined on

G / H

$G/H$ ; moreover, the map

T : f \mapsto F

$T\colon f \mapsto F$ is a bounded linear operator

T : L_{1} (G) \to L_{1} (G / H)

$T\colon L_{1}(G) \to L_{1}(G/H)$ and

\begin{matrix} (2.4) & \int_{G} f d m_{G} = \int_{G / H} F (\hat{x}) d m_{G / H} (\hat{x}) . \end{matrix}

$\begin{equation} \int_{G}f\,dm_{G}=\int_{G/H}F(\widehat{x})\,dm_{G/H}(\widehat{x}). \end{equation} \tag{2.4}$

Since

f_{0} \in L_{2}^{0} (G)

$f_{0} \in L_{2}^{0}(G)$ and

G

$G$ is a compact group, it follows that

f_{0} \in L_{1} (G)

$f_{0} \in L_{1}(G)$ and the function

F_{0} = T f_{0} \in L_{1} (G / H)

$F_{0}=Tf_{0} \in L_{1}(G/H)$ is well defined. Using equality (2.4) for the functions

f_{0}

$f_{0}$ ,

| f_{0} |^{2} \in L_{1} (G)

$|f_{0}|^{2} \in L_{1}(G)$ , we obtain

\begin{aligned} \int_{G / H} F_{0} (\hat{x}) d m_{G / H} (\hat{x}) = \int_{G} f_{0} d m_{G} = 0, \\ \int_{G / H} | F_{0} (\hat{x}) |^{2} d m_{G / H} (\hat{x}) = \int_{G / H} | \int_{H} f_{0} (x + y) d m_{H} (y) |^{2} d m_{G / H} (\hat{x}) \\ ⩽ \int_{G / H} \int_{H} | f_{0} (x + y) |^{2} d m_{H} (y) d m_{G / H} (\hat{x}) \\ = \int_{G} | f_{0} (x) |^{2} d m_{G} (x) = ‖ f_{0} ‖_{L_{2} (G)}^{2} < \infty, \end{aligned}

$\begin{equation*} \begin{aligned} \, &\int_{G/H}F_{0}(\widehat{x})\,dm_{G/H}(\widehat{x})=\int_{G}f_{0}\,dm_{G}=0, \\ &\int_{G/H}|F_{0}(\widehat{x})|^{2}\,dm_{G/H}(\widehat{x})=\int_{G/H} \biggl| \int_{H}f_{0}(x+y)\,dm_{H}(y) \biggr|^{2}\,dm_{G/H}(\widehat{x}) \\ &\qquad \leqslant \int_{G/H} \int_{H} |f_{0}(x+y)|^{2}\,dm_{H}(y)\,dm_{G/H}(\widehat{x}) \\ &\qquad= \int_{G}|f_{0}(x)|^{2}\,dm_{G}(x)=\| f_{0} \|_{L_{2}(G)}^{2} < \infty, \end{aligned} \end{equation*} \notag$

that is,

F_{0} \in L_{2}^{0} (G / H)

$F_{0} \in L_{2}^{0}(G/H)$ .

Now we show that the sums of shifts of $F_{0}$ are dense in the space $L_{2}^{0}(G/H)$ . We fix $P \in L_{2}^{0}(G/H)$ and define a function $p$ on $G$ by $p(g):=P(\widehat{g})$ . Clearly, $P=Tp$ . By Theorem 3 in [9], Ch. VIII, § 39, $p$ is a function in $L_{1}(G)$ , and we have

\int_{G} p d m_{G} = \int_{G / H} P d m_{G / H} = 0, \int_{G} | p |^{2} d m_{G} = \int_{G / H} | P |^{2} d m_{G / H} < \infty .

$\begin{equation*} \int_{G}p\,dm_{G}=\int_{G/H} P\,dm_{G/H}=0, \qquad \int_{G}|p|^{2}\,dm_{G}= \int_{G/H} |P|^{2}\,dm_{G/H} < \infty . \end{equation*} \notag$

Thus,

p \in L_{2}^{0} (G)

$p \in L_{2}^{0}(G)$ , and for each

ε > 0

$\varepsilon>0$ there exist by assumption

n \in N

$n \in \mathbb{N}$ and

g_{k} \in G

$g_{k} \in G$ ,

k = 1, \dots, n

$k=1,\dots ,n$ , such that

\begin{matrix} (2.5) & ‖ p (g) - \sum_{k = 1}^{n} f_{0} (g + g_{k}) ‖_{L_{2} (G)} < ε . \end{matrix}

$\begin{equation} \biggl\| p(g) - \sum_{k=1}^{n}f_{0}(g+g_{k}) \biggr\|_{L_{2}(G)} < \varepsilon. \end{equation} \tag{2.5}$

Using the definition of

F_{0}

$F_{0}$ , equalities (2.3) and (2.4) for the function

| p (\cdot) - \sum_{k = 1}^{n} f_{0} (\cdot + g_{k}) |^{2}

$|p(\,\cdot\,) - \sum_{k=1}^{n}f_{0}(\,\cdot+g_{k})|^{2}$ and inequality (2.5) we obtain

\begin{aligned} ‖ P (\hat{g}) - \sum_{k = 1}^{n} F_{0} (\hat{g} + \hat{g_{k}}) ‖_{L_{2} (G / H)}^{2} \\ = \int_{G / H} | \int_{H} (p (g + y) - \sum_{k = 1}^{n} f_{0} (g + g_{k} + y)) d m_{H} (y) |^{2} d m_{G / H} (\hat{g}) \\ ⩽ \int_{G / H} (\int_{H} | p (g + y) - \sum_{k = 1}^{n} f_{0} (g + g_{k} + y) |^{2} d m_{H} (y)) d m_{G / H} (\hat{g}) \\ = \int_{G} | p (g) - \sum_{k = 1}^{n} f_{0} (g + g_{k}) |^{2} d m_{G} (g) < ε^{2} . \end{aligned}

$\begin{equation*} \begin{aligned} \, &\biggl\| P(\widehat{g}) - \sum_{k=1}^{n}F_{0}(\widehat{g}+\widehat{g_{k}}) \biggr\|_{L_{2}(G/H)}^{2} \\ &\qquad =\int_{G/H} \biggl| \int_{H} \biggl( p(g+y) - \sum_{k=1}^{n}f_{0}(g+g_{k}+y) \biggr)\,dm_{H}(y) \biggr|^{2}\,dm_{G/H}(\widehat{g}) \\ &\qquad \leqslant \int_{G/H} \biggl( \int_{H} \biggl| p(g+y) - \sum_{k=1}^{n}f_{0}(g+g_{k}+y) \biggr|^{2}\,dm_{H}(y) \biggr)\,dm_{G/H}(\widehat{g}) \\ &\qquad =\int_{G} \biggl| p(g) - \sum_{k=1}^{n}f_{0}(g+g_{k}) \biggr|^{2}\,dm_{G}(g) < \varepsilon^{2}. \end{aligned} \end{equation*} \notag$

This means exactly that the sums of shifts of

F_{0}

$F_{0}$ are dense in

L_{2}^{0} (G / H)

$L_{2}^{0}(G/H)$ .

Lemma 2 is proved.

Let $G^{*}$ denote the group of continuous characters of $G$ , and $\mathbb{I}$ denote the unit character on $G$ . Let $H$ be a closed subgroup of the locally compact Abelian group $G$ and $H^{\perp}$ be the set of $\gamma \in G^{*}$ such that $\gamma(h)=1$ for all $h \in H$ . Then $H^{\perp}$ is called the annihilator of $H$ . By [8], Ch. 2, § 2.1.1, $H^{\perp}$ is a closed subgroup of $G^{*}$ .

Lemma 3. Let $G$ be a disconnected compact Abelian group. Then there does not exist a function $f$ in the real space $L_{2}^{0}(G)$ such that the sums of shifts (1.1) of $f$ are dense in this space.

Proof. By [7], Ch. 6, § 24, Theorem 24.25, the character group

G^{*}

$G^{*}$ of a disconnected compact Abelian group

G

$G$ has torsion: it contains a nontrivial element

χ_{0} \in G^{*}

$\chi_{0} \in G^{*}$ of finite order

n_{0} ⩾ 2

$n_{0} \geqslant 2$ . Therefore,

Ξ := {χ_{0}, χ_{0}^{2}, \dots, χ_{0}^{n_{0}} \equiv I}

$\Xi :=\{\chi_{0}, \chi_{0}^{2}, \dots , \chi_{0}^{n_{0}} \equiv \mathbb{I} \}$ is a closed subgroup of

G^{*}

$G^{*}$ of order

n_{0}

$n_{0}$ . Let

H = Ξ^{⊥}

$H=\Xi^{\perp}$ be the annihilator of

Ξ \subset G^{*}

$\Xi \subset G^{*}$ . Then

H

$H$ is a closed subgroup of

(G^{*})^{*}

$(G^{*})^{*}$ , so that by Pontryagin’s duality theorem (see [8], Ch. 1, § 1.7.2)

H

$H$ is a closed subgroup of

G

$G$ and the quotient group

G / H

$G/H$ coincides with

(G^{*})^{*} / Ξ^{⊥}

$(G^{*})^{*}/\,\Xi^{\perp}$ . By [8], Ch. 2, § 2.1.2, the quotient group

(G^{*})^{*} / Ξ^{⊥}

$(G^{*})^{*}/\,\Xi^{\perp}$ is topologically isomorphic to

Ξ^{*}

$\Xi^{*}$ . By [7], Ch. 6, § 23.27.d, the character group

Ξ^{*}

$\Xi^{*}$ of the finite Abelian group

Ξ

$\Xi$ is topologically isomorphic to

Ξ

$\Xi$ . Thus, the disconnected compact Abelian group

G

$G$ contains a closed subgroup

H

$H$ such that

G / H

$G/H$ is topologically isomorphic to a finite group

Ξ

$\Xi$ of order

n_{0} ⩾ 2

$n_{0} \geqslant 2$ .

Assume that there exists a function $f$ in the real space $L_{2}^{0}(G)$ such that the sums (1.1) are dense in this space. Then by Lemma 2 there exists a function $F$ in the real space $L_{2}^{0}(G/H)$ such that the sums of shifts of $F$ are dense in this space. However, $G/H$ is a nontrivial finite Abelian group. This is in contradiction to Lemma 1.

Lemma 3 is proved.

Lemma 4. Let $G$ be a nontrivial compact Abelian group such that its character group $G^{*}$ is not infinite countable. Then in the real space $L_{2}^{0}(G)$ there exists no function $f$ such that the sums of shifts (1.1) are dense in this space.

Proof. If

G^{*}

$G^{*}$ is finite, then by [7], Ch. 6, § 23.27.d,

G

$G$ is topologically isomorphic to

G^{*}

$G^{*}$ , so

G

$G$ is a nontrivial finite Abelian group and the required result follows from Lemma 1.

Consider the case when $G^{*}$ is uncountable. Let $f \in L_{2}^{0}(G)$ . Then by the completeness of the system of characters of a compact Abelian group ([10], Ch. III, § 2, Theorem 3.9) the character group $G^{*}$ is an orthonormal basis of $L_{2}(G)$ ; in particular, $f$ expands in a Fourier series in the system of characters:

f (g) = \sum_{α} c_{α} χ_{α} (g), G^{*} = {χ_{α}},

$\begin{equation*} f(g)=\sum_{\alpha}c_{\alpha}\chi_{\alpha}(g), \qquad G^{*}=\{\chi_{\alpha} \}, \end{equation*} \notag$

where the set of nonzero coefficients

c_{α}

$c_{\alpha}$ is at most countable (otherwise Parseval’s identity does not hold). Hence

f (g) = \sum_{k \in N} c_{α_{k}} χ_{α_{k}} (g), c_{α_{k}} \neq 0,

$\begin{equation*} f(g)=\sum_{k \in \mathbb{N}}c_{\alpha_{k}}\chi_{\alpha_{k}}(g), \qquad c_{\alpha_{k}} \neq 0, \end{equation*} \notag$

and

f (g + h) = \sum_{k \in N} c_{α_{k}} χ_{α_{k}} (h) χ_{α_{k}} (g) .

$\begin{equation*} f(g+h)=\sum_{k \in \mathbb{N}}c_{\alpha_{k}}\chi_{\alpha_{k}}(h)\chi_{\alpha_{k}}(g). \end{equation*} \notag$

Then sums of shifts of

f

$f$ lie in the closed subspace

L

$L$ of the real space

L_{2}^{0} (G)

$L_{2}^{0}(G)$ that is spanned by the functions

χ_{α_{k}}

$\chi_{\alpha_{k}}$ ,

k \in N

$k \in \mathbb{N}$ , and, as a basis of

L_{2}^{0} (G)

$L_{2}^{0}(G)$ is uncountable,

L

$L$ does not coincide with

L_{2}^{0} (G)

$L_{2}^{0}(G)$ .

Lemma 4 is proved.

§ 3. Proof of Theorem 1

Proof. Necessity. This follows from Lemmas 3 and 4.

Sufficiency. Now let $G$ be a connected compact Abelian group with infinite countable character group $G^{*}$ . We prove in several steps that the required function exists.

1. Let $\chi$ be a continuous character on the group $G$ , $\mathbf{0}$ be the identity element of $G$ , and $\mathbb{I}$ be the neutral element of $G^{*}$ . Then $\overline{\chi}$ is also a continuous character on $G$ , and $\chi \equiv \overline{\chi}$ on $G$ if and only if $\chi$ takes only the values $\pm 1$ in $G$ . However, $\chi(\mathbf{0})=1$ and $G$ is a connected group, so $\chi \equiv \overline{\chi}$ if and only if $\chi \equiv \mathbb{I}$ . Thus, the group $G^{*}$ has the form

G^{*} = {χ_{ν}}_{ν = 1}^{\infty} ⊔ {{\bar{χ}}_{ν}}_{ν = 1}^{\infty} ⊔ {I} .

$\begin{equation*} G^{*}=\{\chi_{\nu} \}_{\nu=1}^{\infty} \sqcup \{\overline{\chi}_{\nu} \}_{\nu=1}^{\infty} \sqcup \{\mathbb{I} \}. \end{equation*} \notag$

Note that

G^{*}

$G^{*}$ has a discrete topology (see [8], Ch. 1, § 2, Theorem 1.2.5), and compact subsets of

G^{*}

$G^{*}$ are merely finite subsets of it. By the completeness of the system of characters of a compact Abelian group (see [10], Ch. III, § 2, Theorem 3.9)

G^{*}

$G^{*}$ forms an orthonormal basis of

L_{2} (G)

$L_{2}(G)$ , and each real function

f \in L_{2}^{0} (G)

$f \in L_{2}^{0}(G)$ expands in a Fourier series in the system

G^{*}

$G^{*}$ :

f (g) = \sum_{ν = 1}^{\infty} c_{ν} χ_{ν} (g) + \sum_{ν = 1}^{\infty} \bar{c_{ν}} {\bar{χ}}_{ν} (g), c_{ν} \in C .

$\begin{equation*} f(g)=\sum_{\nu=1}^{\infty}c_{\nu}\chi_{\nu}(g)+\sum_{\nu=1}^{\infty}\overline{c_{\nu}} \overline{\chi}_{\nu}(g), \qquad c_{\nu} \in \mathbb{C}. \end{equation*} \notag$

We seek a function

f

$f$ such that the sums of shifts of

f

$f$ are dense in

L_{2}^{0} (G)

$L_{2}^{0}(G)$ in the following form:

\begin{matrix} (3.1) & f (g) = \sum_{ν = 1}^{\infty} c_{ν} (χ_{ν} (g) + {\bar{χ}}_{ν} (g)), c_{ν} \in R . \end{matrix}

$\begin{equation} f(g)=\sum_{\nu=1}^{\infty}c_{\nu}(\chi_{\nu}(g)+\overline{\chi}_{\nu}(g)), \qquad c_{\nu} \in \mathbb{R}. \end{equation} \tag{3.1}$

2. By [7], Ch. 6, § 24, Theorem 24.15, the topological weight $\mu(G)$ of the compact Abelian group $G$ coincides with the cardinality of $G^{*}$ , that is, it is infinite countable. By [7], Ch. 6, § 25, Theorem 25.14, if the topological weight $\mu(G)$ of a connected compact Abelian group $G$ does not exceed the cardinality of a continuum, then $G$ is monothetic, that is, there exists $g_{0} \in G$ such that $\overline{\{ng_{0}\colon n \in \mathbb{Z} \}}=G$ . By [7], Ch. 6, § 25, Theorem 25.11, each nontrivial character is distinct from 1 at the element $g_{0}$ : $\chi_{\nu}(g_{0}) \neq 1$ , $\nu \in \mathbb{N}$ . Therefore,

\begin{matrix} (3.2) & \forall k \in N \exists δ_{k} > 0 : | χ_{ν} (g_{0}) - 1 | ⩾ δ_{k} for ν = 1, \dots, k . \end{matrix}

$\begin{equation} \forall\, k \in \mathbb{N} \quad \exists\, \delta_{k}>0\colon \quad|\chi_{\nu}(g_{0}) - 1| \geqslant \delta_{k} \quad \text{for } \nu=1,\dots,k. \end{equation} \tag{3.2}$

For each

k \in N

$k \in \mathbb{N}$ fix

ε_{k} \in (0, δ_{k} / k)

$\varepsilon_{k} \in (0, \delta_{k}/k)$ . By Dirichlet’s theorem on simultaneous approximation (see [11], Ch. 1, § 5) there exists a sequence of positive integers

{N_{k}}_{k = 0}^{\infty}

$\{N_{k} \}_{k=0}^{\infty}$ such that

\begin{matrix} (3.3) & N_{0} = 1, N_{k} ⩾ k N_{k - 1} and | (χ_{ν} (g_{0}))^{N_{k}} - 1 | < ε_{k}, ν = 1, \dots, k, k \in N; \end{matrix}

$\begin{equation} N_{0}=1, \qquad N_{k} \geqslant kN_{k-1} \quad \text{and} \quad |(\chi_{\nu}(g_{0}))^{N_{k}} - 1| < \varepsilon_{k}, \qquad \nu=1,\dots,k, \quad k \in \mathbb{N}; \end{equation} \tag{3.3}$

in particular, the sequence

{N_{k}}_{k = 0}^{\infty}

$\{N_{k} \}_{k=0}^{\infty}$ satisfies the condition

\begin{matrix} (3.4) & N_{k + m} ⩾ (k + m) (k + m - 1) \dots (k + 1) N_{k}, k, m \in N . \end{matrix}

$\begin{equation} N_{k+m} \geqslant (k+m)(k+m-1)\dotsb(k+1)N_{k}, \qquad k, m \in \mathbb{N}. \end{equation} \tag{3.4}$

3. By [7], Ch 6, § 25, Theorem 25.18, 2 $\Rightarrow$ 3, given a connected compact Abelian group $G$ , there exists a homomorphism $\varphi\colon G^{*} \to \mathbb{R}_{d}$ into the additive group $\mathbb{R}_{d}$ of real numbers endowed with the discrete topology. According to another part of the same result (see [7], Ch. 6, § 25, Theorem 25.18, 3 $\Rightarrow$ 1), $G$ is solenoidal, that is, there exists a continuous homomorphism $\tau \colon \mathbb{R} \to G$ such that $\overline{\tau(\mathbb{R})}=G$ , and we can see from the proof that

\begin{matrix} (3.5) & \forall χ \in G^{*}, \forall t \in R : χ (τ (t)) = \exp (i t φ (χ)) . \end{matrix}

$\begin{equation} \forall\, \chi \in G^{*}, \quad \forall\, t \in \mathbb{R}\colon\quad \chi(\tau(t))= \exp(it\varphi(\chi)). \end{equation} \tag{3.5}$

Set

\begin{matrix} (3.6) & a_{m} := min {\frac{1}{2^{m}}, \frac{1}{(m^{2} | φ (χ_{m}) |)}}, m \in N . \end{matrix}

$\begin{equation} a_{m} :=\min \biggl\{\frac{1}{2^{m}}, \frac{1}{(m^{2}|\varphi(\chi_{m})|)}\biggr \}, \qquad m \in \mathbb{N}. \end{equation} \tag{3.6}$

Here

φ (χ_{m}) \neq 0

$\varphi(\chi_{m}) \ne 0$ because otherwise

χ_{m} \equiv 1

$\chi_{m} \equiv 1$ by identity (3.5) since the image of

τ

$\tau$ is dense in

G

$G$ .

4. We show that the function

\begin{matrix} (3.7) & ρ (g, h) := \sum_{m = 1}^{\infty} a_{m} | χ_{m} (g) - χ_{m} (h) |, g, h \in G, \end{matrix}

$\begin{equation} \rho(g, h) :=\sum_{m=1}^{\infty}a_{m} |\chi_{m}(g) - \chi_{m}(h)|, \qquad g, h \in G, \end{equation} \tag{3.7}$

is a metric on

G

$G$ . The function

ρ

$\rho$ is well defined because equalities (3.6) imply the estimate

ρ (g, h) ⩽ \sum_{m = 1}^{\infty} 1 / 2^{m - 1} = 2

$\rho(g,h) \leqslant \sum_{m=1}^{\infty} 1/2^{m-1}=2$ for

g, h \in G

$g, h \in G$ . Clearly,

ρ

$\rho$ is nonnegative, symmetric and, by the triangle inequality for the modulus, satisfies the triangle inequality. If

ρ (g, h) = 0

$\rho(g, h)=0$ , then

χ_{m} (g) = χ_{m} (h)

$\chi_{m}(g)=\chi_{m}(h)$ for each

m \in N

$m \in \mathbb{N}$ , so that

(α (g - h)) (χ) = χ (g - h) = 1

$(\alpha(g- h))(\chi)=\chi(g-h)=1$ for all

χ \in G^{*}

$\chi \in G^{*}$ , where

α : G \to (G^{*})^{*}

$\alpha\colon G \to (G^{*})^{*}$ is the canonical isomorphism ([8], Ch. 1, §§ 1.7.1–1.7.2). Therefore,

α (g - h)

$\alpha(g-h)$ is the identity element of the group

(G^{*})^{*}

$(G^{*})^{*}$ , and so

g - h = 0

$g-h= \mathbf{0}$ . It follows from (3.7) that

\begin{matrix} (3.8) & | χ_{m} (g) - χ_{m} (h) | = | {\bar{χ}}_{m} (g) - {\bar{χ}}_{m} (h) | ⩽ \frac{ρ (g, h)}{a_{m}}, m \in N, \end{matrix}

$\begin{equation} |\chi_{m}(g) - \chi_{m}(h)|=|\overline{\chi}_{m}(g) - \overline{\chi}_{m}(h)| \leqslant \frac{\rho(g,h)}{a_{m}}, \qquad m \in \mathbb{N}, \end{equation} \tag{3.8}$

so that each character

χ \in G^{*}

$\chi \in G^{*}$ is a Lipschitz function with respect to

ρ

$\rho$ . Moreover, it is obvious from the definition of

ρ

$\rho$ that this metric is shift invariant.

Now we show that the topology on $G$ induced by $\rho$ coincides with the topology of the group $G$ . By [8], Ch. 1, § 1.2.6, and [8], Ch. 1, § 1.7.2, a basis of topology on $G$ consists of the sets

N (x, C, r) = {y \in G : | γ (y) - γ (x) | < r for all γ \in C},

$\begin{equation*} N(x, C, r)=\{y \in G\colon |\gamma(y)-\gamma(x)|<r\text{ for all }\gamma \in C \}, \end{equation*} \notag$

where

x \in G

$x \in G$ ,

C

$C$ is a compact subset of

G^{*}

$G^{*}$ and

r > 0

$r>0$ . First we show that for all

ε > 0

$\varepsilon>0$ and

x \in G

$x \in G$ there exist a compact set

C \subset G^{*}

$C \subset G^{*}$ and

r > 0

$r>0$ such that

N (x, C, r) \subset B_{ε} (x) := {y \in G : ρ (x, y) < ε}

$N(x, C, r) \subset B_{\varepsilon}(x) :=\{y \in G\colon\rho(x, y) < \varepsilon \}$ . We choose

M \in N

$M \in \mathbb{N}$ such that

1 / 2^{M} < ε / 4

${1/2^{M} < \varepsilon/4}$ , and set

r := ε / 2

$r:= \varepsilon/2$ and

C := {χ_{1}, \dots, χ_{M}}

$C:=\{\chi_{1},\dots ,\chi_{M} \}$ . Then for all

y \in N (x, C, r)

$y \in N(x, C, r)$ and

m = 1, \dots, M

$m=1,\dots,M$ we have the inequality

| χ_{m} (y) - χ_{m} (x) | < r

$|\chi_{m}(y)-\chi_{m}(x)|<r$ , so that by the definition of the coefficients

a_{m}

$a_{m}$ ,

m \in N

$m \in \mathbb{N}$ ,

ρ (x, y) ⩽ \sum_{m = 1}^{M} a_{m} | χ_{m} (y) - χ_{m} (x) | + \sum_{m = M + 1}^{\infty} \frac{1}{2^{m - 1}} < \sum_{m = 1}^{M} \frac{r}{2^{m}} + \frac{1}{2^{M - 1}} < ε,

$\begin{equation*} \rho(x, y) \leqslant \sum_{m=1}^{M} a_{m} |\chi_{m}(y) - \chi_{m}(x)|+\sum_{m=M+1}^{\infty} \frac{1}{2^{m-1}} < \sum_{m=1}^{M} \frac{r}{2^{m}}+\frac{1}{2^{M-1}} < \varepsilon, \end{equation*} \notag$

that is,

y \in B_{ε} (x)

$y \in B_{\varepsilon}(x)$ .

Next we show that for all $r>0$ and $x \in G$ and each compact set $C \subset G^{*}$ there exists $\varepsilon>0$ such that $B_{\varepsilon}(x) \subset N(x, C, r)$ . Because $G^{*}$ is infinite countable and $C \subset G^{*}$ is a compact set, $C$ is finite and there exists $M \in \mathbb{N}$ such that $C \subset \{\mathbb{I}, \chi_{1}, \overline{\chi}_{1}, \dots , \chi_{M}, \overline{\chi}_{M} \}$ . Set $\varepsilon:=r\min_{m=1, \dots , M}a_{m}$ . If $y \in B_{\varepsilon}(x)$ , then by (3.8) we have $|\chi_{m}(y) - \chi_{m}(x)|=|\overline{\chi}_{m}(y) - \overline{\chi}_{m}(x)| < \varepsilon/a_{m} \leqslant r$ for $m=1,\dots , M$ , that is, $y \in N(x, C, r)$ .

We see that the metric $\rho$ agrees with the topology of the group $G$ .

5. Now we prove that the continuous homomorphism $\tau\colon \mathbb{R} \to G$ defined in part 3 of the proof is Lipschitz with respect to the metric $\rho$ on $G$ . Let $u, v \in \mathbb{R}$ . Then from (3.5) and (3.7) we obtain

\begin{aligned} ρ (τ (u), τ (v)) & = \sum_{m = 1}^{\infty} a_{m} | \exp (i u φ (χ_{m})) - \exp (i v φ (χ_{m})) | \\ ⩽ | u - v | (\sum_{m = 1}^{\infty} a_{m} | φ (χ_{m}) |) ⩽ | u - v | (\sum_{m = 1}^{\infty} \frac{1}{m^{2}}) ⩽ 2 | u - v |, \end{aligned}

$\begin{equation*} \begin{aligned} \, \rho(\tau(u), \tau(v)) &=\sum_{m=1}^{\infty}a_{m} |\exp(iu\varphi(\chi_{m})) -\exp(iv\varphi(\chi_{m}))| \\ &\leqslant |u-v| \biggl( \sum_{m=1}^{\infty} a_{m}|\varphi(\chi_{m})| \biggr) \leqslant |u-v| \biggl( \sum_{m=1}^{\infty} \frac{1}{m^{2}} \biggr) \leqslant 2|u-v|, \end{aligned} \end{equation*} \notag$

where in the penultimate inequality we used the definition (3.6) of the

a_{m}

$a_{m}$ .

6. For each $\nu \in \mathbb{N}$ we choose a constant $c_{\nu}$ so that

\begin{matrix} (3.9) & 0 < c_{ν} < min {\frac{1}{N_{ν}}, \frac{a_{ν}}{ν}}, \end{matrix}

$\begin{equation} 0 < c_{\nu} < \min \biggl\{\frac{1}{N_{\nu}},\frac{a_{\nu}}{\nu} \biggr\}, \end{equation} \tag{3.9}$

where

N_{ν}

$N_{\nu}$ and

a_{ν}

$a_{\nu}$ were defined in parts 2 and 3 of the proof, respectively. Then the function

f

$f$ is defined by (3.1). Using inequalities (3.4) and (3.9) we can estimate the norm of

f

$f$ in

L_{2} (G)

$L_{2}(G)$ :

‖ f ‖_{2}^{2} = 2 \sum_{ν = 1}^{\infty} | c_{ν} |^{2} < 2 \sum_{ν = 1}^{\infty} \frac{1}{N_{ν}^{2}} ⩽ 2 \sum_{ν = 1}^{\infty} \frac{1}{N_{1}^{2} (ν!)^{2}} ⩽ 4.

$\begin{equation*} \|f\|_{2}^{2}= 2\sum_{\nu=1}^{\infty}|c_{\nu}|^{2} < 2\sum_{\nu=1}^{\infty}\frac{1}{N_{\nu}^{2}} \leqslant 2 \sum_{\nu=1}^{\infty} \frac{1}{N_{1}^{2}(\nu !)^{2}} \leqslant 4. \end{equation*} \notag$

Hence

f \in L_{2}^{0} (G)

$f \in L_{2}^{0}(G)$ . Next we estimate the following norm:

\begin{aligned} ‖ \sum_{m = 0}^{N_{k} - 1} f (x + m g_{0}) ‖_{2}^{2} = ‖ \sum_{m = 0}^{N_{k} - 1} \sum_{ν = 1}^{\infty} (c_{ν} χ_{ν} (x) (χ_{ν} (g_{0}))^{m} + c_{ν} {\bar{χ}}_{ν} (x) ({\bar{χ}}_{ν} (g_{0}))^{m}) ‖_{2}^{2} \\ = ‖ \sum_{ν = 1}^{\infty} c_{ν} (\sum_{m = 0}^{N_{k} - 1} (χ_{ν} (g_{0}))^{m}) χ_{ν} (x) + \sum_{ν = 1}^{\infty} c_{ν} (\sum_{m = 0}^{N_{k} - 1} ({\bar{χ}}_{ν} (g_{0}))^{m}) {\bar{χ}}_{ν} (x) ‖_{2}^{2} \\ = 2 \sum_{ν = 1}^{\infty} | c_{ν} \sum_{m = 0}^{N_{k} - 1} (χ_{ν} (g_{0}))^{m} |^{2} ⩽ 2 \sum_{ν = 1}^{k} | c_{ν} |^{2} | \frac{(χ_{ν} (g_{0}))^{N_{k}} - 1}{χ_{ν} (g_{0}) - 1} |^{2} + 2 \sum_{ν = k + 1}^{\infty} | c_{ν} |^{2} N_{k}^{2} . \end{aligned}

$\begin{equation*} \begin{aligned} \, &\biggl\|\sum_{m=0}^{N_{k}-1} f(x+mg_{0}) \biggr\|_{2}^{2}=\biggl\| \sum_{m=0}^{N_{k}-1} \sum_{\nu=1}^{\infty} \bigl( c_{\nu} \chi_{\nu}(x) (\chi_{\nu}(g_{0}))^{m}+c_{\nu} \overline{\chi}_{\nu}(x) (\overline{\chi}_{\nu}(g_{0}))^{m} \bigr) \biggr\|_{2}^{2} \\ &\qquad =\biggl\| \sum_{\nu=1}^{\infty} c_{\nu} \biggl( \sum_{m=0}^{N_{k}-1} (\chi_{\nu}(g_{0}))^{m} \biggr) \chi_{\nu}(x)+\sum_{\nu=1}^{\infty} c_{\nu} \biggl( \sum_{m=0}^{N_{k}-1} (\overline{\chi}_{\nu}(g_{0}))^{m} \biggr) \overline{\chi}_{\nu}(x) \biggr\|_{2}^{2} \\ &\qquad =2 \sum_{\nu=1}^{\infty} \biggl| c_{\nu} \sum_{m=0}^{N_{k}-1} (\chi_{\nu}(g_{0}))^{m} \biggr|^{2} \leqslant 2 \sum_{\nu=1}^{k} |c_{\nu}|^{2} \biggl| \frac{(\chi_{\nu}(g_{0}))^{N_{k}} - 1}{\chi_{\nu}(g_{0}) - 1} \biggr|^{2}+2 \sum_{\nu=k+1}^{\infty}|c_{\nu}|^{2} N_{k}^{2}. \end{aligned} \end{equation*} \notag$

Using conditions (3.2)–(3.4) and (3.9) and the definition of the constants

ε_{k}

$\varepsilon_{k}$ we obtain

\begin{aligned} ‖ \sum_{m = 0}^{N_{k} - 1} f (x + m g_{0}) ‖_{2}^{2} ⩽ 2 \sum_{ν = 1}^{k} \frac{1}{N_{ν}^{2}} (\frac{ε_{k}}{δ_{k}})^{2} + 2 \sum_{ν = k + 1}^{\infty} \frac{N_{k}^{2}}{N_{ν}^{2}} \\ ⩽ \frac{2}{k^{2}} \sum_{ν = 1}^{k} \frac{1}{N_{ν}^{2}} + 2 \sum_{ν = k + 1}^{\infty} \frac{N_{k}^{2}}{ν^{2} (ν - 1)^{2} \dots (k + 1)^{2} N_{k}^{2}} \\ ⩽ \frac{2}{k^{2}} k + \frac{2}{(k + 1)^{2}} (1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \dots) ⩽ \frac{2}{k} + \frac{4}{(k + 1)^{2}} \to 0, k \to \infty . \end{aligned}

$\begin{equation*} \begin{aligned} \, &\biggl\|\sum_{m=0}^{N_{k}-1} f(x+mg_{0}) \biggr\|_{2}^{2} \leqslant 2 \sum_{\nu=1}^{k} \frac{1}{N_{\nu}^{2}} \biggl( \frac{\varepsilon_{k}}{\delta_{k}} \biggr)^{2}+2 \sum_{\nu=k+1}^{\infty} \frac{N_{k}^{2}}{N_{\nu}^{2}} \\ &\qquad\leqslant \frac{2}{k^{2}} \sum_{\nu=1}^{k} \frac{1}{N_{\nu}^{2}}+2 \sum_{\nu=k+1}^{\infty} \frac{N_{k}^{2}}{\nu^{2} (\nu -1)^{2}\dotsb (k+1)^{2}N_{k}^{2}} \\ &\qquad \leqslant \frac{2}{k^{2}}k+\frac{2}{(k+1)^{2}}\biggl( 1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+ \dotsb\biggr) \leqslant \frac{2}{k}+\frac{4}{(k+1)^{2}} \to 0, \qquad k \to \infty. \end{aligned} \end{equation*} \notag$

Hence

- f \in S

$-f \in S$ , where

S = \bar{{\sum_{k = 1}^{n} f (x + h_{k}), h_{k} \in G, n \in N}}

$\begin{equation*} S=\overline{\biggl\{\sum_{k=1}^{n} f(x+h_{k}),\,h_{k} \in G,\,n \in \mathbb{N} \biggr\}} \end{equation*} \notag$

(the closure in

L_{2} (G)

$L_{2}(G)$ ). Therefore,

- f (\cdot + h)

$-f(\,\cdot+h)$ belongs to

S

$S$ for all

h \in G

$h \in G$ , that is,

S

$S$ is a closed additive subgroup of

L_{2}^{0} (G)

$L_{2}^{0}(G)$ .

7. We require the following result.

Lemma A ([12], Lemma 4). Let $S$ be a closed additive subgroup of a uniformly smooth Banach space $X$ with modulus of smoothness $s(t)$ , $t \geqslant 0$ . If $a, b \in S$ and for each $\varepsilon > 0$ there exist $x_{0},\dots ,x_{n} \in S$ such that $x_{0}=a$ , $x_{n}=b$ and $\sum_{k=1}^{n}s(\|x_{k}-x_{k-1}\|) < \varepsilon$ , then the whole line segment $[a, b]$ lies in $S$ .

Let $h \in G$ and $\varepsilon>0$ . Since $\tau\colon \mathbb{R} \to G$ is a homomorphism with dense image and the metric $\rho$ agrees with the topology of $G$ (see parts 3 and 4 of the proof), there exists $w \in \mathbb{R}$ such that $\rho(\tau(w), h)< \sqrt{\varepsilon/2}$ . Fix an integer $N > 1+8|w|^{2}/\varepsilon$ and set

h_{k} := τ (\frac{k w}{N - 1}), k = 0, \dots, N - 1, h_{N} := h .

$\begin{equation*} h_{k}:=\tau \biggl( \frac{kw}{N-1} \biggr), \qquad k=0,\dots ,N-1, \quad h_{N}:=h. \end{equation*} \notag$

As the metric

ρ

$\rho$ is translation invariant and

τ

$\tau$ is a bi-Lipschitz homomorphism (see parts 4 and 5 of the proof), we obtain the estimate

\begin{aligned} \sum_{k = 1}^{N} (ρ (h_{k - 1}, h_{k}))^{2} = \sum_{k = 1}^{N - 1} (ρ (τ (\frac{(k - 1) w}{N - 1}), τ (\frac{k w}{N - 1})))^{2} + (ρ (τ (w), h))^{2} \\ < (N - 1) (ρ (τ (0), τ (\frac{w}{N - 1})))^{2} + \frac{ε}{2} ⩽ (N - 1) \cdot 4 | \frac{w}{N - 1} |^{2} + \frac{ε}{2} < ε . \end{aligned}

$\begin{equation*} \begin{aligned} \, & \sum_{k=1}^{N}(\rho(h_{k-1}, h_{k}))^{2}=\sum_{k=1}^{N-1} \biggl( \rho \biggl( \tau \biggl( \frac{(k-1)w}{N-1} \biggr), \tau \biggl(\frac{kw}{N-1} \biggr) \biggr) \biggr)^{2}+ (\rho(\tau(w), h))^{2} \\ &\qquad < (N-1)\biggl( \rho \biggl( \tau(0), \tau \biggl( \frac{w}{N-1} \biggr) \biggr) \biggr)^2+ \frac{\varepsilon}{2} \leqslant (N-1)\cdot 4 \biggl|\frac{w}{N-1} \biggr|^{2}+ \frac{\varepsilon}{2} < \varepsilon. \end{aligned} \end{equation*} \notag$

We can estimate the sum

\sum_{k = 1}^{N} ‖ f (x + h_{k}) - f (x + h_{k - 1}) ‖_{2}^{2}

$\begin{equation*} \sum_{k=1}^{N} \bigl\|f(x+h_{k}) - f(x+h_{k-1}) \bigr\|_{2}^{2} \end{equation*} \notag$

by using the above bound and inequalities (3.8) and (3.9):

\begin{matrix} (3.10) & \begin{aligned} \sum_{k = 1}^{N} ‖ f (x + h_{k}) - f (x + h_{k - 1}) ‖_{2}^{2} \\ = \sum_{k = 1}^{N} ‖ \sum_{ν = 1}^{\infty} (c_{ν} (χ_{ν} (h_{k}) - χ_{ν} (h_{k - 1})) χ_{ν} (x) + c_{ν} ({\bar{χ}}_{ν} (h_{k}) - {\bar{χ}}_{ν} (h_{k - 1})) {\bar{χ}}_{ν} (x)) ‖_{2}^{2} \\ = 2 \sum_{k = 1}^{N} \sum_{ν = 1}^{\infty} | c_{ν} (χ_{ν} (h_{k}) - χ_{ν} (h_{k - 1})) |^{2} ⩽ 2 (\sum_{ν = 1}^{\infty} \frac{c_{ν}^{2}}{a_{ν}^{2}}) (\sum_{k = 1}^{N} (ρ (h_{k - 1}, h_{k}))^{2}) \\ < 2 ε \sum_{ν = 1}^{\infty} \frac{1}{ν^{2}} < 4 ε . \end{aligned} \end{matrix}

$\begin{equation} \begin{aligned} \, \notag &\sum_{k=1}^{N} \bigl\|f(x+h_{k}) - f(x+h_{k-1}) \bigr\|_{2}^{2} \\ \notag &\qquad =\sum_{k=1}^{N} \biggl\| \sum_{\nu=1}^{\infty} (c_{\nu}(\chi_{\nu}(h_{k}) - \chi_{\nu}(h_{k-1})) \chi_{\nu}(x)+c_{\nu}(\overline{\chi}_{\nu}(h_{k}) - \overline{\chi}_{\nu}(h_{k-1})) \overline{\chi}_{\nu}(x)) \biggr\|_{2}^{2} \\ \notag &\qquad =2 \sum_{k=1}^{N} \sum_{\nu=1}^{\infty} |c_{\nu}(\chi_{\nu}(h_{k}) - \chi_{\nu}(h_{k-1}))|^{2} \leqslant 2 \biggl( \sum_{\nu=1}^{\infty} \frac{c_{\nu}^{2}}{a_{\nu}^{2}} \biggr) \biggl( \sum_{k=1}^{N}(\rho(h_{k-1}, h_{k}))^{2} \biggr) \\ &\qquad < 2 \varepsilon \sum_{\nu=1}^{\infty} \frac{1}{\nu^{2}} < 4\varepsilon. \end{aligned} \end{equation} \tag{3.10}$

Thus, the subgroup $S$ defined in part 6 of the proof lies in the space $L_{2}^{0}(G)$ with modulus of smoothness $s(t)=\sqrt{1+t^{2}} - 1=O(t^{2})$ (for instance, see [13], Ch. 1, § e) and moreover, the functions $f(x+h_{0})=f(x)$ , $f(x+h_{1})$ , $\dots$ , $f(x+h_{N})=f(x+h)$ belong to $S$ and (3.10) holds. Then by Lemma A, for each $\lambda \in [0,1]$ the function $\lambda f(x)+(1- \lambda) f(x+h)$ belongs to $S$ . Hence for each $\lambda \in \mathbb{R}$ we also have $\lambda (f(x)- f(x+h)) \in S$ . Therefore, $S$ contains the closed $\mathbb{R}$ -linear subspace $L$ spanned by the functions of the form $f(\,\cdot\,) - f(\cdot+ h)$ , $h \in G$ .

8. We show that $L$ coincides with the real space $L_{2}^{0}(G)$ . Otherwise there exists a nontrivial real function $r \in L_{2}^{0}(G)$ such that

\int_{G} (f (x + h) - f (x)) r (x) d m (x) \equiv 0 ⟹ \int_{G} f (x + h) r (x) d m (x) \equiv c o n s t, h \in G .

$\begin{equation*} \int_{G}(f(x+h)-f(x))r(x)\,dm(x) \equiv 0 \quad\!\Longrightarrow\!\quad \int_{G}f(x+h)r(x)\,dm(x) \equiv \mathrm{const}, \quad h \in G. \end{equation*} \notag$

r

$r$ is real valued, its Fourier expansion looks like

r (x) = \sum_{ν = 1}^{\infty} d_{ν} χ_{ν} (x) + \sum_{ν = 1}^{\infty} \bar{d_{ν}} {\bar{χ}}_{ν} (x), d_{ν} \in C .

$\begin{equation*} r(x)=\sum_{\nu=1}^{\infty}d_{\nu} \chi_{\nu}(x)+\sum_{\nu=1}^{\infty} \overline{d_{\nu}} \overline{\chi}_{\nu}(x), \qquad d_{\nu} \in \mathbb{C}. \end{equation*} \notag$

Now by the expansion

f (x + h) = \sum_{ν = 1}^{\infty} c_{ν} χ_{ν} (h) χ_{ν} (x) + \sum_{ν = 1}^{\infty} c_{ν} {\bar{χ}}_{ν} (h) {\bar{χ}}_{ν} (x)

$\begin{equation*} f(x+h)=\sum_{\nu=1}^{\infty}c_{\nu}\chi_{\nu}(h)\chi_{\nu}(x)+ \sum_{\nu=1}^{\infty}c_{\nu}\overline{\chi}_{\nu}(h)\overline{\chi}_{\nu}(x) \end{equation*} \notag$

we have

\sum_{ν = 1}^{\infty} c_{ν} \bar{d_{ν}} χ_{ν} (h) + \sum_{ν = 1}^{\infty} c_{ν} d_{ν} {\bar{χ}}_{ν} (h) \equiv c o n s t, h \in G .

$\begin{equation*} \sum_{\nu=1}^{\infty}c_{\nu}\overline{d_{\nu}}\chi_{\nu}(h)+ \sum_{\nu=1}^{\infty}c_{\nu}d_{\nu}\overline{\chi}_{\nu}(h) \equiv \mathrm{const}, \qquad h \in G. \end{equation*} \notag$

Since the sequences

{c_{ν}}_{ν \in N}

$\{c_{\nu} \}_{\nu \in \mathbb{N}}$ and

{d_{ν}}_{ν \in N}

$\{d_{\nu} \}_{\nu \in \mathbb{N}}$ belong to

l_{2}

$l_{2}$ , the left-hand side of the above identity is an absolutely convergent Fourier series in

h

$h$ , and therefore

c_{ν} d_{ν} = c_{ν} \bar{d_{ν}} = 0

$c_{\nu}d_{\nu}= c_{\nu}\overline{d_{\nu}}=0$ ,

ν \in N

$\nu \in \mathbb{N}$ . Since

c_{ν} > 0

$c_{\nu} > 0$ for

ν \in N

$\nu \in \mathbb{N}$ , we obtain

d_{ν} = 0

$d_{\nu}=0$ for

ν \in N

$\nu \in \mathbb{N}$ , that is,

r \equiv 0

$r\equiv 0$ , which contradicts the assumptions.

Thus, the subspace $L$ , and therefore the subgroup $S$ , coincides with the real space $L_{2}^{0}(G)$ .

Theorem 1 is proved.

§ 4. Complex case

Remark 1. Let $G$ be a nontrivial compact Abelian group. Then there does not exist a function $f$ in the complex space $L_{2}^{0}(G)$ whose sums of shifts (1.1) are dense in this space.

In fact, given a disconnected compact Abelian group $G$ or a nontrivial compact Abelian group $G$ whose character group $G^{*}$ is not infinite countable, if there exists a function $f$ such that the sums of shifts of $f$ are dense in the complex space $L_{2}^{0}(G)$ , then the sums of shifts of $\operatorname{Re} f$ are dense in the real space $L_{2}^{0}(G)$ , in contradiction to Lemma 3 or Lemma 4, respectively.

Now let $G$ be a connected compact Abelian group with infinite countable character group $G^{*}=\Gamma_{1} \sqcup \Gamma_{2} \sqcup \{\mathbb{I} \}$ , where $\Gamma_{1}=\{\chi_{\nu} \}_{\nu=1}^{\infty}$ and $\Gamma_{2}=\{\overline{\chi}_{\nu} \}_{\nu=1}^{\infty}$ , and assume that there exists a function $f$ in the complex space $L_{2}^{0}(G)$ such that the sums of shifts of $f$ are dense in this space. Let $m_{G}$ and $m_{G^{*}}$ be the Haar measures on $G$ and $G^{*}$ , respectively. Then the Fourier transform

g (χ) := \hat{f} (χ) = \int_{G} f (x) χ (- x) d m_{G} (x)

$\begin{equation*} g(\chi):=\widehat{f}(\chi)=\int_{G}f(x)\chi(-x)\,dm_{G}(x) \end{equation*} \notag$

f

$f$ and also the function

g_{1} : G^{*} \to C, g_{1} (χ) := {\begin{cases} g (\bar{χ}), & χ \in Γ_{1}, \\ - g (\bar{χ}), & χ \in Γ_{2}, \\ 0, & χ \equiv I, \end{cases}

$\begin{equation*} g_{1}\colon G^{*} \to \mathbb{C}, \qquad g_{1}(\chi) := \begin{cases} g(\overline{\chi}), &\chi \in \Gamma_{1}, \\ -g(\overline{\chi}), &\chi \in \Gamma_{2}, \\ 0, &\chi \equiv \mathbb{I}, \end{cases} \end{equation*} \notag$

belong to

L_{2} (G^{*})

$L_{2}(G^{*})$ (see [8], Ch. 1, vol. 1.6.1), and the inverse Fourier transform

f_{1} := \overset{ˇ}{g_{1}}

$f_{1}:=\check{g_{1}}$ belongs to

L_{2}^{0} (G)

$L_{2}^{0}(G)$ because

\int_{G} f_{1} (x) d m_{G} (x) = \hat{f_{1}} (I) = g_{1} (I) = 0.

$\begin{equation*} \int_{G}f_{1}(x)dm_{G}(x)= \widehat{f_{1}}(\mathbb{I})=g_{1}(\mathbb{I})=0. \end{equation*} \notag$

Note that by assumption the function

f

$f$ is nontrivial, so that

g, g_{1}

$g, g_{1}$ and

f_{1}

$f_{1}$ are also nontrivial. For

y \in G

$y \in G$ we have

\begin{aligned} \int_{G} f (x + y) \bar{f_{1} (x)} d m_{G} (x) = \int_{G^{*}} \hat{f (\cdot + y)} (χ) \bar{\hat{f_{1} (\cdot)} (χ)} d m_{G^{*}} (χ) \\ = \int_{G^{*}} χ (y) \hat{f} (χ) \bar{g_{1} (χ)} d m_{G^{*}} (χ) = \int_{G^{*}} χ (y) g (χ) \bar{g_{1} (χ)} d m_{G^{*}} (χ) \\ = \int_{Γ_{1}} χ (y) g (χ) \bar{g (\bar{χ})} d m_{G^{*}} (χ) - \int_{Γ_{2}} χ (y) g (χ) \bar{g (\bar{χ})} d m_{G^{*}} (χ) \\ = \int_{Γ_{1}} χ (y) g (χ) \bar{g (\bar{χ})} d m_{G^{*}} (χ) - \int_{Γ_{1}} \bar{χ} (y) g (\bar{χ}) \bar{g (χ)} d m_{G^{*}} (χ) \\ = \int_{Γ_{1}} (χ (y) g (χ) \bar{g (\bar{χ})} - \bar{χ (y) g (χ) \bar{g (\bar{χ})}}) d m_{G^{*}} (χ) . \end{aligned}

$\begin{equation*} \begin{aligned} \, &\int_{G} f(x+y)\overline{f_{1}(x)}\,dm_{G}(x)=\int_{G^{*}} \widehat{f(\cdot+y)}(\chi) \overline{\widehat{f_{1}(\,\cdot\,)}(\chi)}\,dm_{G^{*}}(\chi) \\ &\qquad =\int_{G^{*}} \chi(y) \widehat{f}(\chi) \overline{g_{1}(\chi)}\,dm_{G^{*}}(\chi) =\int_{G^{*}} \chi(y) g(\chi) \overline{g_{1}(\chi)}\,dm_{G^{*}}(\chi) \\ &\qquad =\int_{\Gamma_{1}} \chi(y) g(\chi) \overline{g(\overline{\chi})}\,dm_{G^{*}}(\chi) - \int_{\Gamma_{2}} \chi(y) g(\chi) \overline{g(\overline{\chi})}\,dm_{G^{*}}(\chi) \\ &\qquad =\int_{\Gamma_{1}} \chi(y) g(\chi) \overline{g(\overline{\chi})}\,dm_{G^{*}}(\chi) - \int_{\Gamma_{1}} \overline{\chi}(y) g(\overline{\chi}) \overline{g(\chi)}\,dm_{G^{*}}(\chi) \\ &\qquad =\int_{\Gamma_{1}} \bigl( \chi(y) g(\chi) \overline{g(\overline{\chi})} - \overline{\chi(y) g(\chi) \overline{g(\overline{\chi})}} \bigr)\,dm_{G^{*}}(\chi). \end{aligned} \end{equation*} \notag$

Thus,

Re \int_{G} f (x + y) \bar{f_{1} (x)} d m_{G} (x) = 0

$\begin{equation*} \operatorname{Re} \int_{G} f(x+y)\overline{f_{1}(x)}\,dm_{G}(x)=0 \end{equation*} \notag$

for all

y \in G

$y \in G$ , and thus sums of shifts of

f

$f$ lie in a real hyperplane in the complex space

L_{2}^{0} (G)

$L_{2}^{0}(G)$ , so that they cannot be dense in this space.

The author is grateful to P. A. Borodin for stating the problem and making useful comments.



Bibliography

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Citation: N. A. Dyuzhina, “Density of the sums of shifts of a single function in the

L_{2}^{0}

$L_2^0$ space on a compact Abelian group”, Sb. Math., 215:6 (2024), 743–754

Citation in format AMSBIB

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\by N.~A.~Dyuzhina

\paper Density of the sums of shifts of a~single function in the $L_2^0$ space on a~compact Abelian group

\jour Sb. Math.

\yr 2024

\vol 215

\issue 6

\pages 743--754

\mathnet{http://mi.mathnet.ru/eng/sm10011}

\crossref{https://doi.org/10.4213/sm10011e}

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This publication is cited in the following 1 articles:

Yu. A. Skvortsov, “Plotnost summ sdvigov odnoi funktsii dlya deistviya kompaktnoi gruppy”, Matem. zametki, 117:3 (2025), 479–482

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