Matrix-free interpolation on the sphere

摘要

We study a subspace of bivariate trigonometric polynomials for interpolating functions on the sphere. We give an explicit construction for a system of interpolation nodes, and the corresponding basis for this space, that allows a ( discrete) fast Fourier transform-type formula for the interpolant. We prove that the uniform norm of our interpolation operator is of the order (log M)(2), where M is the number of interpolation points. We also construct a minimal quadrature rule for our space ( with a number of points equal to the dimension of the space), and describe an associated interpolation operator.