On a generalization of the Bessel function Neumann expansion

TL;DR

This paper generalizes the Bessel-Neumann expansion by introducing a broader class of basis functions satisfying differential equations, demonstrating potential for faster convergence in function representation.

Contribution

It proposes a new expansion framework using basis functions governed by differential equations, extending the classical Bessel function expansion.

Findings

Non-standard basis functions can achieve faster convergence.

A procedure for computing basis functions and coefficients is developed.

Theoretical properties of the generalized expansion are analyzed.

Abstract

The Bessel-Neumann expansion (of integer order) of a function $g:\mathbb{C}\rightarrow\mathbb{C}$ corresponds to representing $g$ as a linear combination of basis functions $\phi_0,\phi_1,\ldots$, i.e., $g(z)=\sum_{\ell = 0}^\infty w_\ell \phi_\ell(s)$, where $\phi_i(z)=J_i(z)$, $i=0,\ldots$, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.