Related papers: Fourier series with the continuous primitive integ…
We study the arithmetic Fourier transforms of trace functions on general connected commutative algebraic groups. To do so, we first prove a generic vanishing theorem for twists of perverse sheaves by characters, and using this tool, we…
Let $S_{n}f$ denote the $n$th partial sum of the Vilenkin-Fourier series of a function $f \in L^{1}(G)$. For $1 < p_{-} \leq p_{+} < \infty$, we characterize all exponents $p(\cdot)$ for which the convergence of $S_{n}f$ to $f$ in…
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…
We continue the study of the Hrushovski-Kazhdan integration theory and consider exponential integrals. The Grothendieck ring is enlarged via a tautological additive character and hence can receive such integrals. We then define the Fourier…
In the present article the author extends the Fourier transform to a more general class of functions; First to power-law functions with integer and half-integer exponents then to the widely used quantum statistics function (Fermi-Dirac and…
Let $p$ be an integer $\geq2$ and let $K$ be a global field. A foliated $p$-adic F-series is a function $X$ of a $p$-adic integer variable $\mathfrak{z}$ satisfying the functional equations…
We construct functions in the disc algebra with pointwise universal Fourier series on sets which are G-delta and dense and at the same time with Fourier series whose set of divergence is of Hausdorff dimension zero. We also see that some…
We connect through the Fourier transform shift-invariant Sobolev type spaces $V_s\subset H^s$, $s\in\mathbb R,$ and the spaces of periodic distributions and analyze the properties of elements in such spaces with respect to the product. If…
Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of…
Fourier-Dedekind sums are a generalization of Dedekind sums - important number-theoretical objects that arise in many areas of mathematics, including lattice point enumeration, signature defects of manifolds and pseudo random number…
The Arithmetic Fourier Transform is a numerical formulation for computing Fourier series and Taylor series coefficients. It competes with the Fast Fourier Transform in terms of speed and efficiency, requiring only addition operations and…
Most of the known Fourier transforms associated with the equations of mathematical physics have a trivial kernel, and an inversion formula as well as the Parseval equality are fulfilled. In other words, the system of the eigenfunctions…
In this paper, we deal with the convolution series that are a far reaching generalization of the conventional power series and the power series with the fractional exponents including the Mittag-Leffler type functions. Special attention is…
We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\mathbb T^m$ such that the sequence of the Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m), ~1\leq…
For the multiple Fourier series of the periodization of some radial functions on $\mathbb{R}^d$, we investigate the behavior of the spherical partial sum. We show the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third…
Liu and Liu introduced the random Fourier transform, which is a random Fourier series in Hermite functions, and applied it to image encryption and decryption. They expected its applications in optics and information technology. These…
In this paper we generalize Bochkariev's theorem, which states that for any uniformly bounded orthonormal system $\Phi$, there exists a Lebesgue integrable function such that the Fourier series of it with respect to system $\Phi$ diverge on…
Sawin recently gave an axiomatic characterization of multiple Dirichlet series over the function field $\mathbb{F}_{q}(T)$ and proved their existence by exhibiting the coefficients as trace functions of specific perverse sheaves. However,…
The convergence of DP Fourier series which are neither strongly convergent nor strongly divergent is discussed in terms of the Taylor series of the corresponding inner analytic functions. These are the cases in which the maximum disk of…
The ring of periodic distributions on ${\mathbb{R}}^{\tt d}$ with usual addition and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring ${\mathcal{S}}'({\mathbb{Z}}^{\tt d})$ of all maps…