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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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

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

of shifts of

$f$ :

(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)?

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.

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

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

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

$N \geqslant 2$ . Each

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

$m$ , and

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

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

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

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

$L$ of vectors of length

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

$f$ has mean value zero, for each

$g \in G$ we have

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

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

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

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

From (2.1) and (2.2) we obtain

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

If the vectors

$f_{g_{n}}$ ,

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

$S$ , and therefore also its closure

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

$\overline{S}$ cannot coincide with the

$(N-1)$ -dimensional space

$L$ . If the vectors

$f_{g_{n}}$ ,

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

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

$N-2$ , so it cannot coincide with

$L$ of dimension

$N-1$ .

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

$G$ is a compact Abelian group and

$H$ is a closed subgroup of it,

$G/H$ 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

$x \in G$ by the subgroup

$H$ by

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

$G, H$ and

$G/H$ are endowed with the Haar measures

$m_{G}$ ,

$m_{H}$ and

$m_{G/H}$ such that

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

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

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

is well defined on

$G/H$ ; moreover, the map

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

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

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

Since

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

$G$ is a compact group, it follows that

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

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

$f_{0}$ ,

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

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

that is,

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

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

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

Thus,

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

$\varepsilon>0$ there exist by assumption

$n \in \mathbb{N}$ and

$g_{k} \in G$ ,

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

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

Using the definition of

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

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

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

This means exactly that the sums of shifts of

$F_{0}$ are dense in

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

Lemma 2 is proved.

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

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

$G$ has torsion: it contains a nontrivial element

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

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

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

$G^{*}$ of order

$n_{0}$ . Let

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

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

$H$ is a closed subgroup of

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

$H$ is a closed subgroup of

$G$ and the quotient group

$G/H$ coincides with

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

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

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

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

$\Xi$ is topologically isomorphic to

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

$G$ contains a closed subgroup

$H$ such that

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

$\Xi$ of order

$n_{0} \geqslant 2$ .

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

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

$G$ is topologically isomorphic to

$G^{*}$ , so

$G$ 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:

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

where the set of nonzero coefficients

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

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

and

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

Then sums of shifts of

$f$ lie in the closed subspace

$L$ of the real space

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

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

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

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

$L$ does not coincide with

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

Lemma 4 is proved.

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

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

Note that

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

$G^{*}$ 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)

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

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

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

$G^{*}$ :

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

We seek a function

$f$ such that the sums of shifts of

$f$ are dense in

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

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

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,

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

For each

$k \in \mathbb{N}$ fix

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

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

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

in particular, the sequence

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

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

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

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

Set

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

Here

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

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

$\tau$ is dense in

$G$ .

4. We show that the function

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

is a metric on

$G$ . The function

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

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

$g, h \in G$ . Clearly,

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

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

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

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

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

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

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

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

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

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

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

so that each character

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

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

$\rho$ 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

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

where

$x \in G$ ,

$C$ is a compact subset of

$G^{*}$ and

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

$\varepsilon>0$ and

$x \in G$ there exist a compact set

$C \subset G^{*}$ and

$r>0$ such that

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

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

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

$r:= \varepsilon/2$ and

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

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

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

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

$a_{m}$ ,

$m \in \mathbb{N}$ ,

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

that is,

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

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

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

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

$a_{m}$ .

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

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

where

$N_{\nu}$ and

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

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

$f$ in

$L_{2}(G)$ :

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

Hence

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

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

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

$\varepsilon_{k}$ we obtain

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

Hence

$-f \in S$ , where

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

(the closure in

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

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

$S$ for all

$h \in G$ , that is,

$S$ is a closed additive subgroup of

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

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 $S$ .

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

\Bibitem{Dyu24}

\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}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4804036}

\zmath{https://zbmath.org/?q=an:07945693}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2024SbMat.215..743D}

\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001334620600002}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85206913915}

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

§ 1. Introduction

§ 2. Auxiliary lemmas

§ 3. Proof of Theorem 1

§ 4. Complex case

Bibliography