Product formulas and convolution structure for Fourier-Bessel series

Abstract

One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter\(\alpha = n + \frac{1}{2}\),n=0, 1, 2,⋯, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases\(\alpha = \frac{3}{2}\) and\(\alpha = \frac{5}{2}\). As a consequence, a positive convolution structure is established for\(\alpha \in \{ \frac{1}{2},\frac{3}{2},\frac{5}{2}\}\). The method of proof is based on solving a hyperbolic initial boundary value problem.