Fractional Zernike functions

TL;DR

This paper studies fractional Zernike functions on the punctured unit disc, extending classical Zernike polynomials, and explores their spectral properties, orthogonality, differential equations, and applications in Hilbert space decompositions.

Contribution

It introduces and analyzes fractional Zernike functions, establishing their spectral realization, orthogonality, and connections to special functions and differential operators.

Findings

Fractional Zernike functions are orthogonal eigenfunctions of a perturbed magnetic Laplacian.

They have explicit integral representations and generating functions.

A subclass forms a complete orthogonal system in a Hilbert space.

Abstract

We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated $\beta$-restricted Zernike functions. Mainly, we give the spectral realization of the latter ones and show that they are orthogonal $L^2$-eigenfunctions for certain perturbed magnetic (hyperbolic) Laplacian. The algebraic and analytic properties for the fractional Zernike functions to be established include the connection to special functions, their zeros, their orthogonality property, as well as the differential equations, recurrence, and operational formulas they satisfy. Integral representations are also obtained. Their regularity as poly-meromorphic functions is discussed and their generating functions including a bilinear one of "Hardy--Hille type" are derived. Moreover, we prove that a truncated subclass defines a complete orthogonal system in the underlying Hilbert space giving rise to a specific Hilbertian orthogonal decomposition in terms of a second class of generalized Bergman spaces.