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

A. Koskela, E. Jarlebring

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### Sammanfattning

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.
Originalspråk engelska Numerical Mathematics and Advanced Applications ENUMATH 2017 F. Radu, K. Kumar, I. Berre, J. Nordbotten, I. Pop 10 126 Springer, Cham 5 jan 2019 205-214 978-3-319-96414-0 978-3-319-96415-7 https://doi.org/10.1007/978-3-319-96415-7_17 Publicerad - 5 jan 2019 Ja A4 Artikel i en konferenspublikation European Conference on Numerical Mathematics and Advanced Applications ENUMATH 2017 - Voss, NorgeVaraktighet: 25 sep 2017 → 29 sep 2017

### Publikationsserier

Namn Lecture Notes in Computational Science and Engineering Springer-Verlag 1439-7358

### Vetenskapsgrenar

• 111 Matematik

### Citera det här

Koskela, A., & Jarlebring, E. (2019). On a Generalization of Neumann Series of Bessel Functions Using Hessenberg Matrices and Matrix Exponentials. I F. Radu, K. Kumar, I. Berre, J. Nordbotten, & I. Pop (Red.), Numerical Mathematics and Advanced Applications ENUMATH 2017 (Vol. 126, s. 205-214). (Lecture Notes in Computational Science and Engineering). Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_17