On function spaces for radial functions

2024/03/14 by M. D. Groves, Groves, Mark D., Dan J. Hill +1 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces

paper · pdf · doi:10.48550/arxiv.2403.09372

Abstract

This paper is concerned with complex Banach-space valued functions of the form f_k(rcosθ,rsinθ,z)=e^{i k θ}f_k(r,z), r ∈ [0,∞), θ∈ \mathbbT¹, z ∈ ℝ, for some k ∈ ℤ. It is demonstrated how classical and Sobolev spaces for the radial function f_k can be constructed in a natural fashion from the corresponding standard function spaces for f_k. A theory of radial distributions is derived in the same spirit. Finally, a new class of Hankel spaces for the case f_k=f_k(r) is introduced. These spaces are the radial counterparts of the familiar Bessel-potential spaces for functions defined on ℝ^d. The paper concludes with an application of the theory to the Dirichlet boundary-value problem for Poisson's equation in a cylindrical domain.

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