Related papers: Segal-Bargmann transform and Paley-Wiener theorems…
Usually such area of mathematics as differential equations acts as a consumer of results given by functional analysis. This article will give an example of the reverse interaction of these two fields of knowledge. Namely, the derivation and…
Motivated by the two-parameter free unitary Segal-Bargmann transform in the form of conditional expectation, we derive the integral transform representation of the two-parameter free unitary Segal-Bargmann transform which coincides to the…
A kind of generalized Gelfand pair is introduced via a Banach algebra consisting of bi-invariant functions in a weighted Lebesgue space. The related spherical functions and the Fourier transformation are constructed. The multipliers of the…
In this paper we introduce the classical Segal-Bargmann transform starting from the basis of Hermite polynomials and extend it to Clifford algebra-valued functions. Then we apply the results to monogenic functions and prove that the…
We introduce and discuss some basic properties of some integral transforms in the framework of specific functional Hilbert spaces, the holomorphic Bargmann-Fock spaces on $\mathbb{C}$ and $\mathbb{C}^2$ and the slice hyperholomorphic…
We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, one of which involves a limit transition from Opdam's results for the graded Hecke algebra. Furthermore, the connection…
We study the reproducing kernel for weighted polynomial Bergman spaces and consider applications to the Berezin transform. Some of our results have applications in random matrix theory, a topic which we discuss in a separate (companion)…
We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for H\"ormander's…
The aim of this article is to give an overview of several types of Paley-Wiener theorems occuring in harmonic analysis related to symmetric spaces. This will serve as a motivation for the introduction of the $\Theta$-spherical functions,…
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with a help of a family of weight functions is considered in the paper. For this…
We systematically develop real Paley-Wiener theory for the Fourier transform on R^d for Schwartz functions, L^p-functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions…
We extend the Paley--Wiener theorem for riemannian symmetric spaces to an important class of infinite dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit…
One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then the harmonic…
Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we~obtain expansions for the Szeg\"o and the weighted Bergman kernels of $M$-harmonic functions, i.e.~functions annihilated by the invariant…
We introduce and study weighted spaces of functions with mixed norm on the upper half-plane, defined in terms of Fourier transform. We give a characterization of analytic functions within these spaces, and in particular, we provide an…
We use a variant of the classical Segal-Bargmann transform to understand the canonical quantization of Yang-Mills theory on a space-time cylinder. This transform gives a rigorous way to make sense of the Hamiltonian on the gauge-invariant…
We introduce new classes of right quaternionic Hilbert spaces of Bargmann-Fock type $\mathcal{GB}_{m}^{2}(\mathbb{H})$, labeled by nonnegative integer $m$, generalizing the so-called slice hyperholomorphic Bargmann-Fock space introduced…
We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's…
In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This…
We first prove a Cauchy's integral theorem and Cauchy type formula for certain inhomogeneous Cimmino system from quaternionic analysis perspective. The second part of the paper directs the attention towards some applications of the…