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

A. Koskela, E. Jarlebring

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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.
Titel på värdpublikationNumerical Mathematics and Advanced Applications ENUMATH 2017
RedaktörerF. Radu, K. Kumar, I. Berre, J. Nordbotten, I. Pop
Antal sidor10
FörlagSpringer, Cham
Utgivningsdatum5 jan. 2019
ISBN (tryckt)978-3-319-96414-0
ISBN (elektroniskt)978-3-319-96415-7
StatusPublicerad - 5 jan. 2019
Externt publiceradJa
MoE-publikationstypA4 Artikel i en konferenspublikation
EvenemangEuropean Conference on Numerical Mathematics and Advanced Applications ENUMATH 2017 - Voss, Norge
Varaktighet: 25 sep. 201729 sep. 2017


NamnLecture Notes in Computational Science and Engineering
ISSN (tryckt)1439-7358


  • 111 Matematik

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