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Anton Tselishchev

Publications and source records attributed to Anton Tselishchev.

At least 19 recordsLinked to original sources
arXiv2026

A new example of crystalline-type measure with unit masses and an application to tiling of the real line by translates of a function

In this note we provide a simple explicit example of a discrete set $Λ\subset\mathbb{R}$ of unbounded density such that the Fourier transform of its counting measure $\widehatδ_Λ$ is a tempered distribution of the form $cδ_0'' + μ$ where $μ$ is a discrete measure supported by a union of three arithmetic progressions. This example is then used to construct a translational tiling of unbounded density of the real line by a function with compact support.

math.CA↗
arXiv2026

Gabor unconditional bases and frames in $L^p(\mathbb{R})$

We consider the following problem: given a set $Λ\subset \mathbb{R} \times \mathbb{R}$ and $p \neq 2$, does there exist a function $g \in L^p(\mathbb{R})$ such that the Gabor system $\{g(x-t) e^{2 πisx}\}$, $(t,s) \in Λ$, consisting of time-frequency shifts of $g$, forms an unconditional basis or unconditional Schauder frame in the space $L^p(\mathbb{R})$? We completely resolve this question for $p>2$; in particular, we characterize the sets $Λ$ such that an unconditional Schauder frame of this form exists. We also prove a Balian-Low type result, showing that the window function $g$ cannot enjoy mild continuity and decay conditions. For $1<p<2$, we prove that a Gabor system cannot form an unconditional basis or unconditional Schauder frame in $L^p(\mathbb{R})$ if the set $Λ$ satisfies a natural separation condition.

math.CA↗
arXiv2026

Weighted Chui's conjecture

The goals of this paper are threefold. First, we show that a counterpart of the Newman bound related to the Chui conjecture is valid in the case where the gradient of Coulomb potential is generated by arbitrary positive charges placed at the boundary of a unit ball. Second, we prove that our bound is sharp in the two-dimensional case. Finally, we discuss a related problem, where the unit charges are placed in the unit disc.

math.CA↗
arXiv2026

Completeness of sparse, almost integer and finite local complexity sequences of translates in $L^p(\mathbb{R})$

A real sequence $Λ= \{λ_n\}_{n=1}^\infty$ is called $p$-generating if there exists a function $g$ whose translates $\{g(x-λ_n)\}_{n=1}^\infty$ span the space $L^p(\mathbb{R})$. While the $p$-generating sets were completely characterized for $p=1$ and $p>2$, the case $1 < p \le 2$ remains not well understood. In this case, both the size and the arithmetic structure of the set play an important role. In the present paper, (i) We show that a $p$-generating set $Λ$ of positive real numbers can be very sparse, namely, the ratios $λ_{n+1} / λ_n$ may tend to $1$ arbitrarily slowly; (ii) We prove that every "almost integer" sequence $Λ$, i.e. satisfying $λ_n = n + α_n$, $0 \neq α_n \to 0$, is $p$-generating; and (iii) We construct $p$-generating sets $Λ$ such that the successive differences $λ_{n+1} - λ_n$ attain only two different positive values. The constructions are, in a sense, sharp: it is well known that $Λ$ cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.

math.CA↗
arXiv2025

Schauder frames of discrete translates in $L^2(\mathbb{R})$

We construct a uniformly discrete sequence $\{λ_1 < λ_2 < \cdots\} \subset \mathbb{R}$ and functions $g$ and $\{g_n^*\}$ in $L^2(\mathbb{R})$, such that every $f \in L^2(\mathbb{R})$ admits a series expansion \[ f(x) = \sum_{n=1}^{\infty} \langle f, g_n^* \rangle \, g(x-λ_n) \] convergent in the $L^2(\mathbb{R})$ norm.

math.CA↗
arXiv2025

Schauder frames of discrete translates in $L^p(\mathbb{R})$

For every $p > (1 + \sqrt{5})/2$ we construct a uniformly discrete real sequence $\{λ_n\}_{n=1}^\infty$ satisfying $|λ_n| = n + o(1)$, a function $g \in L^p(\mathbb{R})$, and continuous linear functionals $\{g^*_n\}_{n=1}^\infty$ on $L^p(\mathbb{R})$, such that every $f \in L^p(\mathbb{R})$ admits a series expansion \[ f(x) = \sum_{n=1}^{\infty} g_n^*(f) g(x-λ_n) \] convergent in the $L^p(\mathbb{R})$ norm. We moreover show that $g$ can be chosen nonnegative.

math.CA↗
arXiv2025

Unconditional Schauder frames of exponentials and of uniformly bounded functions in $L^p$ spaces

It is known that there is no unconditional basis of exponentials in the space $L^p(Ω)$, $p \ne 2$, for any set $Ω\subset \mathbb{R}^d$ of finite measure. This is a consequence of a more general result due to Gaposhkin, who proved that the space $L^p(Ω)$ does not admit a seminormalized unconditional basis consisting of uniformly bounded functions. We show that the latter result fails if the word "basis" is replaced with "Schauder frame". On the other hand we prove that if $Ω$ has nonempty interior then there are no unconditional Schauder frames of exponentials in the space $L^p(Ω)$, $p \ne 2$.

math.CA↗
arXiv2025

Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps

We prove that if every element $u$ in a Hilbert space $H$ admits a representation as unconditionally convergent series $$u=\sum_{k=1}^\infty \langle u, y_k\rangle x_k,$$ then there exist nonzero scalars $\{α_k\}_{k=1}^\infty$ such that both sequences $\{α_k x_k\}_{k=1}^\infty$ and $\{\overlineα_k^{-1}y_k\}_{k=1}^\infty$ are frames. Our result has the following equivalent reformulation: if $Φ:\ell^\infty\to B(H)$ is a bounded linear map such that for every element of the unit vector basis $e_k$ in $\ell^\infty$ the operator $Φ(e_k)$ has rank one, then $Φ$ is completely bounded.

math.FA↗
arXiv2024

Reverse Carleson measures for spaces of analytic functions

Let $X$ be a quasi-Banach space of analytic functions in the unit disc and let $q>0$. A finite positive Borel measure $μ$ in the closed unit disc $\overline{\mathbb{D}}$ is called a $q$-reverse Carleson measure for $X$ if and only if there exists a constant $C>0$ such that $$\|f\|_{X}\leq C \|f\|_{L^q(\overline{\mathbb D},dμ)} $$ for all $f\in X\cap C(\overline{\mathbb D})$. We fully characterize the $q$-reverse Carleson measures with all $q>0$ for Hardy spaces $H^p(\mathbb D)$ with all $0<p\leq \infty$, for the space $\mathrm{BMOA}(\mathbb D)$ and for the Bloch space. In addition, we describe $q$-reverse Carleson measures for the holomorphic Triebel--Lizorkin spaces $HF_0^{q,r}$ and the holomorphic Besov spaces $HB_0^{q,r}$. Related results are obtained for the Hardy spaces and certain holomorphic Triebel--Lizorkin spaces in the unit ball of $\mathbb{C}^d$.

math.CV↗
arXiv2024

There are no unconditional Schauder frames of translates in $L^p(\mathbb{R})$, $1 \le p \le 2$

It is known that a system formed by translates of a single function cannot be an unconditional Schauder basis in the space $L^p(\mathbb{R})$ for any $1 \le p < \infty$. To the contrary, there do exist unconditional Schauder frames of translates in $L^p(\mathbb{R})$ for every $p>2$. The existence of such a system for $1 < p \leq 2$, however, has remained an open problem. In this paper the problem is solved in the negative: we prove that none of the spaces $L^p(\mathbb{R})$, $1 \le p \le 2$, admits an unconditional Schauder frame of translates.

math.CA↗
arXiv2024

Schur property for jump parts of gradient measures

We consider weakly null sequences in the Banach space of functions of bounded variation $\mathrm{BV}(\mathbb{R}^d)$. We prove that for any such sequence $\{f_n\}$ the jump parts of the gradients of functions $f_n$ tend to $0$ strongly as measures. It implies that Dunford--Pettis property for the space $\mathrm{SBV}$ is equivalent to the Dunford--Pettis property for the Sobolev space $W^{1,1}.$

math.FA↗
arXiv2024

Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces

For any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n = \infty$ we provide an explicit simple construction of $(1+\varepsilon_n)$-bounded Markushevich basis in a separable Hilbert space which is not strong, or, in other terminology, is not hereditary complete; this condition on the sequence $(\varepsilon_n)_{n=1}^\infty$ is sharp. Using a finite-dimensional version of this construction, Dvoretzky's theorem and a construction of Vershynin, we conclude that in any Banach space for any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n^2 = \infty$ there exists a $(1+\varepsilon_n)$-bounded Markushevich basis which is not a Schauder basis after any permutation of its elements.

math.FA↗
arXiv2023

Littlewood--Paley--Rubio de Francia inequality for unbounded Vilenkin systems

Rubio de Francia proved the one-sided version of Littlewood--Paley inequality for arbitrary intervals. In this paper, we prove the similar inequality in the context of arbitrary Vilenkin systems (that is, for functions on infinite products of cyclic groups). There are no assumptions on the orders of these groups.

math.CA↗
arXiv2023

On multipliers into martingale $SL^\infty$ spaces for arbitrary filtrations

In this paper we study the following problem: for a given bounded positive function $f$ on a filtered probability space can we find another function (a multiplier) $m$, $0\le m\le 1$, such that the function $mf$ is not ``too small'' but its square function is bounded? We explicitly show how to construct such multipliers for the usual martingale square function and for so-called conditional square function. Besides that, we show that for the usual square function more general statement can be obtained by application of a non-constructive abstract correction theorem by S. V. Kislyakov.

math.PR↗
arXiv2022

Bellman function method for general operators on martingales

It is shown that the Bellman function method can be applied to study the $L^p$-norms of general operators on martingales, i.e., of operators that are not necessarily martingale transforms. Informally, we provide a single Bellman-type function that "encodes" the $L^p$-boundedness of "almost all" operators from Gundy's extrapolation theorem. As examples of such operators, we consider the Haar transforms and the operator whose $L^p$-boundedness underlies Rubio de Francia's inequality for the Walsh system.

math.FA↗
arXiv2021

Absence of local unconditional structure in spaces of smooth functions on the torus of arbitrary dimension

Consider a finite collection $\{T_1, \ldots, T_J\}$ of differential operators with constant coefficients on $\mathbb{T}^n$ ($n\geq 2$) and the space of smooth functions generated by this collection, namely, the space of functions $f$ such that $T_j f \in C(\mathbb{T}^n)$, $1\leq j\leq J$. We prove that if there are at least two linearly independent operators among their senior parts (relative to some mixed pattern of homogeneity), then this space does not have local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of $C(S)$

math.FA↗