On a Generalization of Neumann Series of Bessel Functions Using Hessenberg Matrices and Matrix Exponentials

Abstract

The Neumann expansion of Bessel functions (of integer order) of a function $g:\mathbb {C}\rightarrow \mathbb {C}$ corresponds to representing g as a linear combination of basis functions φ ₀, φ ₁, …, i.e., $g(s)=\sum _{\ell = 0}^\infty w_\ell \varphi _\ell (s)$, where φ _i(s) = J _i(s), i = 0, …, 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.