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arXiv · 2403.09372

On function spaces for radial functions

Mark D. Groves·Dan J. Hill
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Abstract

This paper is concerned with complex Banach-space valued functions of the form $$ \hat{f}_k(r\cosθ,r\sinθ,z)=\mathrm{e}^{\mathrm{i} k θ}f_k(r,z), \qquad r \in [0,\infty), θ\in \mathbb{T}^1, z \in \mathbb{R}, $$ for some $k \in \mathbb{Z}$. 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 $\hat{f}_k$. A theory of radial distributions is derived in the same spirit. Finally, a new class of \textit{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 $\mathbb{R}^d$. 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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Mark D. Groves, Dan J. Hill. 2024-08-21. On function spaces for radial functions. https://arxiv.org/abs/2403.09372

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