Polynomial interpolation results in Sobolev spaces
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Polynomial interpolation and hyperinterpolation over general regions
Journal of Approximation Theory
A New Spectral Boundary Integral Collocation Method for Three-Dimensional Potential Problems
SIAM Journal on Numerical Analysis
On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation
SIAM Journal on Numerical Analysis
Matrix-free Interpolation on the Sphere
SIAM Journal on Numerical Analysis
A wideband fast multipole method for the Helmholtz equation in three dimensions
Journal of Computational Physics
Efficient evaluation of highly oscillatory acoustic scattering surface integrals
Journal of Computational and Applied Mathematics
Shifted GMRES for oscillatory integrals
Numerische Mathematik
A fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions
Journal of Computational Physics
Hilbert scales and Sobolev spaces defined by associated Legendre functions
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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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.