Dual of 2D fractional Fourier transform associated to It\^o--Hermite polynomials

TL;DR

This paper introduces dual integral transforms related to the 2D fractional Fourier transform associated with Itô--Hermite polynomials, analyzing their spectral properties, null spaces, ranges, and connections to fractional Hankel transforms.

Contribution

It provides a novel class of dual transforms on the Gaussian Hilbert space, exploring their spectral characteristics and relationships to other fractional integral transforms.

Findings

Identified null spaces and ranges depending on zeros of Itô--Hermite polynomials

Derived explicit singular values and studied compactness and Schatten class membership

Established connections to fractional Hankel transforms

Abstract

A class of integral transforms, on the planar Gaussian Hilbert space with range in the weighted Bergman space on the bi-disk, is defined as the dual transforms of the 2d fractional Fourier transform associated with the Mehler function for It\^o--Hermite polynomials. Some spectral properties of these transforms are investigated. Namely, we study their boundedness and identify their null spaces as well as their ranges. Such identification depends on the zeros set of It\^o--Hermite polynomials. Moreover, the explicit expressions of their singular values are given and compactness and membership in p-Schatten class are studied. The relationship to specific fractional Hankel transforms is also established