Slepian functions on the sphere, generalized Gaussian quadrature rule

Abstract

Denote by K the operator of 'time-band-time' limiting on the surface of the sphere and consider the problem of computing singular vectors of K. This problem can be reduced to a simpler task of computing eigenfunctions of a differential operator, if a differential operator, which commutes with K and has a simple spectrum, can be exhibited. In Grünbaum et al (1982 SIAM J. Appl. Math. 42 941-55) such a second-order differential operator commuting with K on the appropriate subspaces was constructed. In this paper, this algebraic property of commutativity is used to produce an efficient numerical scheme for computing a convenient basis for the space of singular vectors of K. The basis forms an extended Chebyshev system, and a generalized Gaussian quadrature rule for such a basis is presented.

An alternative to Slepian functions on the unit sphere – A space–frequency analysis based on localized spherical polynomials

“…In comparison to other works (cf. [16,27,39]) dealing with spatio-spectral concentration on the unit sphere, we use the multiplication operator M cos θ :…”

Best Approximation in Hardy Spaces and by Polynomials, with Norm Constraints

“…Note that in the annular setting of H 2 (A) (G = A) and for the diagonal situation where I = T ⊂ ∂A coincides with one of the two connected components of ∂A, the Fourier basis provides an example of such Slepian functions in infinite dimension. More generally, such functions have also been studied [24,31,33] when:…”

“…• G is the unit ball in R 3 with I a polar cap contained in ∂G = S, and the Slepian functions are sought among spherical polynomials of prescribed degree N (spherical harmonic basis); their computation however is not so easy for large N and requires additional considerations, some of which are developed in [24,31].…”

Spatially Restricted Integrals in Gradiometric Boundary Value Problems

“…The eigenvectors which maximize this ratio are selected as the orthogonal base functions with a local concentration. Slepian (1983) presented the method for one-dimensional signals but later it was generalized to two-dimensions by Albertella et al (1999), Miranian (2004) and Wiezorek and Simons (2005) and further developed by Simons et al (2006).…”