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

TL;DR

This paper establishes $L^p$ convergence of Fourier series derived from Dirichlet Laplacian eigenfunctions on specific triangular domains, expanding understanding of spectral convergence in geometric settings.

Contribution

It proves $L^p$ norm convergence for Fourier series on three particular triangles, addressing limitations related to Lamé's Theorem and spectral properties.

Findings

Proves $L^p$ convergence for the 45-90-45 triangle.

Establishes convergence for the equilateral triangle.

Analyzes the limitations of the method in relation to Lamé's Theorem.

Abstract

We prove $L^p$ norm convergence for (appropriate truncations of) the Fourier series arising from the Dirichlet Laplacian eigenfunctions on three types of triangular domains in $\mathbb{R}^2$: (i) the 45-90-45 triangle, (ii) the equilateral triangle and (iii) the hemiequilateral triangle (i.e. half an equilateral triangle cut along its height). The limitations of our argument to these three types are discussed in light of Lam\'e's Theorem.