Related papers: $\ell^2$-Linear Independence for the System of Int…

The aim of this paper is to study some properties of left translates of a square integrable function on the Heisenberg group. First, a necessary and sufficient condition for the existence of the canonical dual to a function $\varphi\in…

We consider an aspect of the open problem: Does every square-integrable function on SU(2) have an almost everywhere convergent Fourier series? Let 0 < alpha < 1. We show that to each countable set E in SU(2) there corresponds an…

Let $G$ be a locally compact group. We examine the problem of determining when nonzero functions in $L^2(G)$ have linearly independent translations. In particular, we establish some results for the case when $G$ has an irreducible, square…

The HRT conjecture states that any finite collection of time-frequency shifts of a non-zero square-integrable function on the real line is linearly independent. In this paper, we establish the linear independence of finite systems of…

One of the motivations to state HRT conjecture on the linear independence of finite Gabor systems was the fact that there are linearly dependent Finite Wavelet Systems (FWS). Meanwhile, there are also many examples of linearly independent…

We show that it is possible for a square integrable function on the circle, which is a sum of an almost everywhere convergent series of exponentials with positive frequencies, to not belong to the Hardy space. A consequence in the…

We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere finite function $f(x)$ which is bounded below by an integrable function, then the series is the Fourier series of $f(x)$ if the…

In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two important results of Godement on square integrable positive definite functions to matrix valued square integrable positive definite…

The Fourier series of continuous functions of constant absolute value have interesting properties : according to the main theorems of the article, if the coefficients with positive indexes are square-summable with respect to a certain…

For $S_k$, the space of cusp forms of weight $k$ for the full modular group, we first introduce periods on $S_k$ associated to symmetric square $L$-functions. We then prove that for a fixed natural number $n$, if $k$ is sufficiently large…

For a given rational number $x$ and an integer $s\geq 1$, let us consider a generalized polylogarithmic function, often called the Lerch function, defined by $$\Phi_{s}(x,z)= \sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}\enspace.$$ We prove…

Let $\mu$ be a positive measure on the real line with locally finite support $\Lambda$ and integer masses such that its Fourier transform in the sense of distributions is a purely point measure. An explicit form is found for an entire…

We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {tk} depend on the function being sampled. The use of such sampling in the construction of Lebesgue's integral sums is noted and discussed as…

Let $f \in L^1(\mathbb{R}^2)$ and let $\widehat f$ be its Fourier integral. We study summability of the partial integral $S_{\rho,\mathsf{H}}(x)=\int_{\{\|y\|_\mathsf{H} \le \rho\}} e^{i x\cdot y}\widehat f(y) dy$, where $\|y\|_\mathsf{H}$…

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$,…

Let $\lambda$ be a positive number, and let $(x_j:j\in\mathbb Z)\subset\mathbb R$ be a fixed Riesz-basis sequence, namely, $(x_j)$ is strictly increasing, and the set of functions $\{\mathbb R\ni t\mapsto e^{ix_jt}:j\in\mathbb Z\}$ is a…

We prove that for any 4 points in the plane that belong to 2 parallel lines, there is no linear dependence between the associated time-frequency translates of any nontrivial Schwartz function. If mild Diophantine properties are satisfied,…

We prove that the HRT conjecture holds when the Gabor system consists of a 4-point set in the time-frequency plane and a square-integrable function that is ultimately positive. We also prove the conjecture for Gabor systems generated by an…

We show that for every ergodic and aperiodic probability preserving system, there exists a $\mathbb{Z}$ valued, square integrable function $f$ such that the partial sums process of the time series $\left\{f\circ T^i\right\}_{i=0}^\infty$…

We consider the problem of determining the Fourier integral in the Hilbert space of square integrable functions. Fourier integral is the scalar product of two functions belonging to the Hilbert space of square integrable functions and the…