A special orthogonal complement basis for holomorphic-Hermite functions and associated 1d - and 2d-fractional Fourier transforms

ABSTRACT

In 1990, van Eijndhoven and Meyers provide a special orthonormal basis for the Bargmann Hilbert space consisting of holomorphic Hermite functions. Then it was be natural to look for its orthogonal complement in the underlying $L^2$-Hilbert space. In this paper, we describe the orthogonal complement of this Hilbert space. More precisely, a polyanalytic orthonormal basis is given and the explicit expressions of the corresponding reproducing kernel functions and Segal–Bargmann integral transforms are provided. The obtained basis are then used to provide a non-trivial $1d$- and $2d$-fractional like-Fourier transforms.

KEYWORDS:

• Generalized Bargmann spaces
• polyanalytic functions
• reproducing kernels
• holomorphic Hermite polynomials
• polyanalytic Hermite polynomials
• Segal–Bargmann transform
• fractional like-Fourier transform

AMS CLASSIFICATIONS:

• 42C05
• 33C45
• 46E22
• 65R10
• 42A38
• 42B10

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Allal Ghanmi http://orcid.org/0000-0003-0764-5576