We consider the following problem: given a set Λ⊂ℝ×ℝ and p ≠ 2, does there exist a function g ∈ L^p(ℝ) such that the Gabor system {g(x-t) e^2 πisx}, (t,s) ∈ Λ, consisting of time-frequency shifts of g, forms an unconditional basis or unconditional Schauder frame in the space L^p(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(ℝ) if the set Λ satisfies a natural separation condition.

We construct a uniformly discrete sequence {lambda(1) < lambda(2) < center dot center dot center dot} subset of R and functions g and {g(n)(& lowast;)} in L-2(R), such that every f is an element of L-2(R) admits a series expansion f(x) = Sigma(infinity)(n=1) < f,g(n)(& lowast;)> g(x-lambda(n)) convergent in the L-2(R) norm. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

For every p>(1+root 5)/2, we construct a uniformly discrete real sequence {lambda(n)}(n=1)(infinity) satisfying divided by lambda(n)divided by(infinity)n=1, a function g is an element of L-p(R), and continuous linear functionals {g(n)(& lowast;)}(n=1)(infinity) on L-p(R), such that every f is an element of L-p(R) admits a series expansion f(x)=& sum;(infinity)(n=1)g(n)(& lowast;)(f)g(x-lambda(n)) convergent in the L-p(R) norm. We moreover show that g can be chosen nonnegative.

It is possible to have a packing by translates of a cube that is maximal (i.e. no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness of exponentials on a domain, we pursue the question whether one can have maximal orthogonal sets of exponentials for a cube without them being complete. We prove that this is not possible in dimensions 1 and 2, but is possible in dimensions 3 and higher. We provide several examples of such maximal incomplete sets of exponentials, differing in size, and we raise relevant questions. We also show that even in dimension 1 there are sets which are spectral (i.e. have a complete set of orthogonal exponentials) and yet they also possess maximal incomplete sets of orthogonal exponentials.

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

Abstract A real sequence is called ‐generating if there exists a function whose translates span the space . While the ‐generating sets were completely characterized for and , the case remains not well understood. In this case, both the size and the arithmetic structure of the set play an important role. In this paper, (i) we show that a ‐generating set of positive real numbers can be very sparse, namely, the ratios may tend to 1 arbitrarily slowly; (ii) we prove that every “almost integer” sequence , that is, satisfying , , is ‐generating; and (iii) we construct ‐generating sets such that the successive differences attain only two different positive values. The constructions are, in a sense, sharp: it is well known that cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.

We construct a uniformly discrete sequence {λ_1 < λ_2 < ⋯}⊂ℝ and functions g and {g_n^*} in L^2(ℝ), such that every f ∈ L^2(ℝ) admits a series expansion f(x) = ∑_n=1^∞⟨ f, g_n^* ⟩ g(x-λ_n) convergent in the L^2(ℝ) norm.

A real sequence Λ= {λ_n}_n=1^∞ is called p-generating if there exists a function g whose translates {g(x-λ_n)}_n=1^∞ span the space L^p(ℝ). While the p-generating sets were completely characterized for p=1 and p>2, the case 1 < p ≤ 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 ≠ α_n → 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.

We construct a real sequence λn =1 ∞ satisfying λn = n + o(1), and a Schwartz function f on ℝ, such that for any N the system of translates f(x − λn), n > N, is complete in the space Lp(ℝ) for every p > 1. The same system is also complete in a wider class of Banach function spaces on ℝ.

It is known that a system formed by translates of a single function cannot be an unconditional Schauder basis in the space L-p(R) for any 1 <= p2. The existence of such a system for 1

An $n \times m$ array with nonnegative entries is called doubly stochastic if the sum of its entries at each row is $m$ and at each column is $n$. The set of all $n \times m$ doubly stochastic arrays is a convex polytope with finitely many extremal points. The main result of this paper characterizes the possible sizes of the supports of all extremal $n \times m$ doubly stochastic arrays. In particular we prove that the minimal size of the support of an $n \times m$ doubly stochastic array is $n + m - \gcd(n,m)$. Moreover, for $m=kn+1$ we also characterize the structure of the support of the extremal arrays.

We construct a nonnegative function $g$ and a set $\Lambda = \{\lambda_1 < \lambda_2 < \dots\}$ of positive real numbers such that the system $\{g(x-\lambda_n)\}_{n=1}^\infty$ is complete simultaneously in all $L^p(\mathbb{R})$ spaces, $1 < p < \infty$, and $\Lambda$ satisfies one of the following two additional conditions: either $\Lambda$ is very sparse, namely, the ratios $\lambda_{n+1}/\lambda_n$ tend to $1$ arbitrarily slowly; or the differences $\lambda_{n+1} - \lambda_n$ attain only two different values, so $\Lambda$ has finite local complexity. Both constructions are, in a sense, extreme: it is well known that $\Lambda$ cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.

We use the measurable Hall's theorem due to Cieśla and Sabok to prove that (i) if two measurable sets A,B ⊂ℝ^d of the same measure are bounded remainder sets with respect to a given irrational d-dimensional vector α, then A, B are equidecomposable with measurable pieces using translations from ℤ α+ ℤ^d; and (ii) given a lattice Γ⊂ℝ^m ×ℝ^n with projections p_1 and p_2 onto ℝ^m and ℝ^n respectively, if two cut-and-project sets in ℝ^m obtained from Riemann measurable windows W, W' ⊂ℝ^n are bounded distance equivalent, then W, W' are equidecomposable with measurable pieces using translations from p_2(Γ). We also prove by a different method that for one-dimensional cut-and-project sets the pieces can be chosen Riemann measurable.

It is known that there is no unconditional basis of exponentials in the space $L^p(\Omega)$, $p \ne 2$, for any set $\Omega \subset \mathbb{R}^d$ of finite measure. This is a consequence of a more general result due to Gaposhkin, who proved that the space $L^p(\Omega)$ 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 $\Omega$ has nonempty interior then there are no unconditional Schauder frames of exponentials in the space $L^p(\Omega)$, $p \ne 2$.

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

We consider measurable functions f$f$ on R$\mathbb {R}$ that tile simultaneously by two arithmetic progressions alpha Z$\alpha \mathbb {Z}$ and beta Z$\beta \mathbb {Z}$ at respective tiling levels p$p$ and q$q$. We are interested in two main questions: what are the possible values of the tiling levels p,q$p,q$, and what is the least possible measure of the support of f$f$? We obtain sharp results which show that the answers depend on arithmetic properties of alpha,beta$\alpha , \beta$ and p,q$p,q$, and in particular, on whether the numbers alpha,beta$\alpha , \beta$ are rationally independent or not.

It is known that a system formed by translates of a single function cannot be an unconditional Schauder basis in the space L^p(ℝ) for any 1 ≤ p < ∞. To the contrary, there do exist unconditional Schauder frames of translates in L^p(ℝ) for every p>2. The existence of such a system for 1 < p ≤ 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(ℝ), 1 ≤ p ≤ 2, admits an unconditional Schauder frame of translates.

A set Ω⊂ℝ^d is said to be spectral if the space L^2(Ω ) admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that Ω is spectral if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it was recently proved that the Fuglede conjecture does hold for the class of convex bodies in ℝ^d . The proof was based on a new geometric necessary condition for spectrality, called “weak tiling”. In this paper we study further properties of the weak tiling notion, and present applications to convex bodies, non-convex polytopes, product domains and Cantor sets of positive measure.

It is well known that the functions f \in L^1(\mathbb{R}^d) whose translates along a lattice \Lambda form a tiling, can be completely characterized in terms of the zero set of their Fourier transform. We construct an example of a discrete set \Lambda \subset \mathbb{R} (a small perturbation of the integers) for which no characterization of this kind is possible: there are two functions f, g \in L^1(\mathbb{R}) whose Fourier transforms have the same set of zeros, but such that f + \Lambda is a tiling while g + \Lambda is not.

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space $L^2(\Omega)$.