Remarks on the $L^p$ convergence of Bessel--Fourier series on the disc

TL;DR

This paper discusses the $L^p$ convergence properties of Bessel--Fourier series on the disk, extending previous results to new norm spaces and ranges of $p$, with implications for eigenfunction expansions of the Laplacian.

Contribution

It extends known $L^p$ convergence results of Bessel--Fourier series on the disk to new norm spaces and a broader range of $p$, providing modified convergence criteria.

Findings

Convergence established for $4/3 < p < 4$ in mixed norm spaces.

Modified results achieve $L^p$ convergence for $2 \,\leq p < 4$ in radial-angular decompositions.

Provides a framework for understanding eigenfunction expansions of the Laplacian on planar domains.

Abstract

The $L^p$ convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for $p\neq 2$. After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose eigenfunctions are explicitly computable as products of trigonometric and Bessel functions. We summarise a result of Balodis and C\'ordoba (1999) regarding the $L^p$ convergence of the Bessel--Fourier series in the mixed norm space $L^p_{\mathrm{rad}}(L^2_{\mathrm{ang}})$ on the disk for the range $\tfrac{4}{3}<p<4$. We then describe how to modify their result to obtain $L^p(\mathbb{D}, r\,\mathrm{d}r\,\mathrm{d}t)$ norm convergence in the subspace $L^p_{\mathrm{rad}}(L^q_{\mathrm{ang}})$($\tfrac{1}{p}+\tfrac{1}{q}=1$) for the restricted range ${2\leq p < 4}$.