On a generalization of the Bessel function Neumann expansion
Abstract
The BesselNeumann 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 nonstandard basis functions can give faster convergence than the Bessel functions.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1712.04387
 Bibcode:
 2017arXiv171204387K
 Keywords:

 Mathematics  Numerical Analysis
 EPrint:
 8 pages, 2 figures