Generalized Fourier-Bessel operator and almost-periodic interpolation and approximation

TL;DR

This paper introduces a generalized Fourier-Bessel operator for efficient evaluation, interpolation, and approximation of functions with rotationally symmetric frequency sets, leveraging the structure of the group $SE(2,N)$ for improved computational methods.

Contribution

It develops an abstract factorization theorem for evaluation functions based on the structure of $SE(2,N)$, enabling efficient numerical solutions for interpolation and approximation problems involving almost-periodic functions.

Findings

Established an abstract factorization theorem for evaluation functions.

Leveraged the structure of $SE(2,N)$ for computational efficiency.

Connected the approach to classical problems like FFT in polar coordinates.

Abstract

We consider functions $f$ of two real variables, given as trigonometric functions over a finite set $F$ of frequencies. This set is assumed to be closed under rotations in the frequency plane of angle $\frac{2k\pi}{M}$ for some integer $M$. Firstly, we address the problem of evaluating these functions over a similar finite set $E$ in the space plane and, secondly, we address the problems of interpolating or approximating a function $g$ of two variables by such an $f$ over the grid $E.$ In particular, for this aim, we establish an abstract factorization theorem for the evaluation function, which is a key point for an efficient numerical solution to these problems. This result is based on the very special structure of the group $SE(2,N)$, subgroup of the group $SE(2)$ of motions of the plane corresponding to discrete rotations, which is a maximally almost periodic group. Although the motivation of this paper comes from our previous works on biomimetic image reconstruction and pattern recognition, where these questions appear naturally, this topic is related with several classical problems: the FFT in polar coordinates, the Non Uniform FFT, the evaluation of general trigonometric polynomials, and so on.