On a generalization of the Bessel function Neumann expansion

2017/12/12 by Koskela, Antti, Jarlebring, Elias
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1712.04387

Abstract

The Bessel-Neumann expansion (of integer order) of a function g:ℂ→ℂ corresponds to representing g as a linear combination of basis functions ϕ₀,ϕ₁,…, i.e., g(z)=∑_{ℓ = 0}^∞ w_ℓ ϕ_ℓ(s), where ϕ_i(z)=J_i(z), 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.

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