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

Abstract

The $L^p$ convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for $p\ne 2$. 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 Córdoba (1999) regarding the $L^p$ convergence of the Bessel--Fourier series in the mixed norm space $L_{\mathrm{rad}}^p(L_{\mathrm{ang}}^2)$ on the disk for the range $\frac{4}{3}<p<4$. 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 $L_{\mathrm{rad}}^p(L_{\mathrm{ang}}^q)$($\frac{1}{p}+\frac{1}{q}=1$) for the restricted range $2\le p<4$.

arXiv comment7 pages

References in corpus (1)

• The $L^p$ convergence of Fourier series on triangular domains

Cited by in corpus (1)

• The $L^p$ convergence of Fourier series on triangular domains