On the generalized oblate spheroidal wave functions and application

TL;DR

This paper introduces generalized oblate spheroidal wave functions (GOSWFs), which are eigenfunctions of a finite bilateral Laplace transform, and demonstrates their effectiveness in function approximation and transform inversion.

Contribution

The paper defines GOSWFs as a new complete family of functions generalizing oblate spheroidal wave functions and Jacobi polynomials, with methods for their computation and applications in transform inversion.

Findings

GOSWFs form a complete basis over finite and infinite intervals.

They outperform classical bases in approximating bilateral Laplace bandlimited functions.

Numerical examples validate the theoretical advantages of GOSWFs.

Abstract

In this paper, we introduce a new set of functions, which have the property of the completeness over a finite and infinite intervals. This family of functions, denoted for simplicity GOSWFs, are a generalization of the oblate spheroidal wave functions. They generalize also the Jacobi polynomials in some sens. The GOSWFs are nothing but the eigenfunctions of the finite weighted bilateral Laplace transform $\mathcal{F}_c^{(\alpha,\beta)}.$ We compute this functions by two methods: In the first one we use a differential operator $\mathcal{D}$ which commutes with $\mathcal{F}_c^{(\alpha,\beta)}.$ In the second one we use the Gaussian quadrature method. As an application, we use the GOSWFs to invert the finite bilateral Laplace transform. We also use the GOSWFs to approximate bilateral weighted Laplace bandlimited functions and we show that they are more advantageous then other classical basis of $L^2((-1,1),(1-x)^\alpha(1+x)^\beta)dx$. Finally, we provide the reader by some numerical examples that illustrate the theoretical results.