Related papers: Fourier transform of composed functions
These are notes of a talk based on the work arXiv:1212.3630 joint with A. Aizenbud. Let V be a finite-dimensional vector space over a local field F of characteristic 0. Let f be a function on V of the form $f(x)= \psi (P(x))$, where P is a…
This paper is devoted to the $L^p(\mathbb R)$ theory of the fractional Fourier transform (FRFT) for $1\le p < 2$. In view of the special structure of the FRFT, we study FRFT properties of $L^1$ functions, via the introduction of a suitable…
We consider functions on the lattice generated by the integer powers of $q^2$ for $0<q<1$ and construct the $q$-analog of Fourier transform based on the Jackson integral in the space of distributions on the lattice.
In this paper, a Fourier series in fractional dimensional space is introduced for an arbitrarily periodic function $f(t;\alpha)$. We call it fractional Fourier series of the order $\alpha$. Extending the basis functions of the linear space…
Fourier transformations of several functions of one and two variables are evaluated and then used to derive some integral and series identities. It is shown that certain double Mordell integrals can be reduced to a sum of products of…
In this paper, we develop a continual analog of decomposition over orthogonal bases in spaces generated by equidistant shifts of a single function. By doing so, we obtain an explicit expression for best approximation by spaces of shifts in…
The main aim of the paper is to present a general version of the Fourier Tauberian theorem for monotone functions. This result, together with Berezin's inequality, allows us to obtain a refined version the Li-Yau estimate for the counting…
We introduce two efficient algorithms for computing the partial Fourier transforms in one and two dimensions. Our study is motivated by the wave extrapolation procedure in reflection seismology. In both algorithms, the main idea is to…
We consider different pentagon identities realized by the hyperbolic hypergeometric functions and investigate their degenerations to the level of complex hypergeometric functions. In particular, we show that one of the degenerations yields…
The double exponential formula was introduced for calculating definite integrals with singular point oscillation functions and Fourier-integrals. The double exponential transformation is not only useful for numerical computations but it is…
We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ being fixed and $x>0$, has a power series expansion…
In this study, we introduce and explore a delay differential equation that lends itself to explicit solutions in the Fourier-transformed space. Through the careful alignment of the initial function, we can construct a highly accurate…
Let $H_k$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ for the full modular group $SL(2, \mathbb{Z})$, and let $j\geq 3$ be any fixed integer. For $f\in H_k$, we write $\lambda_{{\rm{sym}^j…
The Howe dual pair (sl(2),O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are…
A function $f:\mathbb{Z}_n \to \mathbb{C}$ can be represented as a linear combination $f(x)=\sum_{\alpha \in \mathbb{Z}_n}\widehat{f}(\alpha) \chi_{\alpha,n}(x)$ where $\widehat{f}$ is the (discrete) Fourier transform of $f$. Clearly, the…
We introduce a general definition of hybrid transforms for constructible functions. These are integral transforms combining Lebesgue integration and Euler calculus. Lebesgue integration gives access to well-studied kernels and to regularity…
It is shown that performing simultaneously two transformations on functions of space and time (for instance a Fourier transform on the space variable and a Laplace transform on the time variable) can be easier than performing them one after…
The Harish-Chandra--Howe local character expansion expresses the characters of reductive, $p$-adic groups in terms of Fourier transforms of nilpotent orbital integrals on their Lie algebras, and Murnaghan--Kirillov theory expresses many…
The special unitary group SU(2) plays a fundamental role in the description of symmetries in quantum mechanics, theoretical physics, and spherical signal processing. In this paper, we address the computational challenges of performing…
We establish upper bounds for shifted moments of modular $L$-functions to a fixed modulus as well as quadratic twists of modular $L$-functions under the generalized Riemann hypothesis. Our results are then used to establish bounds for…