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

A. Koskela, E. Jarlebring

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review


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.
Original languageEnglish
Title of host publicationNumerical Mathematics and Advanced Applications ENUMATH 2017
EditorsF. Radu, K. Kumar, I. Berre, J. Nordbotten, I. Pop
Number of pages10
PublisherSpringer, Cham
Publication date5 Jan 2019
ISBN (Print)978-3-319-96414-0
ISBN (Electronic)978-3-319-96415-7
Publication statusPublished - 5 Jan 2019
Externally publishedYes
MoE publication typeA4 Article in conference proceedings
EventEuropean Conference on Numerical Mathematics and Advanced Applications ENUMATH 2017 - Voss, Norway
Duration: 25 Sep 201729 Sep 2017

Publication series

NameLecture Notes in Computational Science and Engineering
ISSN (Print)1439-7358

Fields of Science

  • 111 Mathematics

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