A generalization of prolate spheroidal functions with more uniform resolution to the triangle

Abstract

Prolate spheroidal functions constitute a one-parameter (α) family of orthogonal functions in the interval. For α = 0, they are the Legendre polynomials. For larger α, the prolate spheroidal functions oscillate more uniformly than the Legendre polynomials, and provide more uniform resolution in the interval. The prolate spheroidal functions can be obtained by adding a zeroth-order term to the Sturm–Liouville equation for the Legendre polynomials. Here, the Sturm–Liouville equation for orthogonal polynomials in the triangle is modified in a similar fashion. The modification maintains the self-adjointness and symmetry properties of the original Sturm–Liouville equation, so that the new eigenfunctions are orthogonal and give spectrally accurate approximations of smooth functions with arbitrary boundary conditions in the triangle. The properties of the new eigenfunctions mimic those in the interval. For larger α, the new eigenfunctions provide more uniform resolution in the triangle.