  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 ${2\leq p < 4}$.
