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

### Jean-Paul Gauthier and Dario Prandi

**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.