Interpolation and cubature approximations and analysis for a class of wideband integrals on the sphere

Abstract

We propose, analyze, and implement interpolatory approximations and Filon-type cubature for efficient and accurate evaluation of a class of wideband generalized Fourier integrals on the sphere. The analysis includes derivation of (i) optimal order Sobolev norm error estimates for an explicit discrete Fourier transform type interpolatory approximation of spherical functions; and (ii) a wavenumber explicit error estimate of the order $\mathcal {O}(\kappa ^{-\ell } N^{-r_{\ell }})$, for $\ell = 0, 1, 2$, where $\kappa$ is the wavenumber, $2N^2$ is the number of interpolation/cubature points on the sphere and $r_{\ell }$ depends on the smoothness of the integrand. Consequently, the cubature is robust for wideband (from very low to very high) frequencies and very efficient for highly-oscillatory integrals because the quality of the high-order approximation (with respect to quadrature points) is further improved as the wavenumber increases. This property is a marked advantage compared to standard cubature that require at least ten points per wavelength per dimension and methods for which asymptotic convergence is known only with respect to the wavenumber subject to stable of computation of quadrature weights. Numerical results in this article demonstrate the optimal order accuracy of the interpolatory approximations and the wideband cubature.