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

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