A spectral interpolation scheme on the unit sphere based on the nodes of spherical Lissajous curves

Abstract

Abstract For sampling values along spherical Lissajous curves we establish a spectral interpolation and quadrature scheme on the sphere. We provide a mathematical analysis of spherical Lissajous curves and study the characteristic properties of their intersection points. Based on a discrete orthogonality structure we are able to prove the unisolvence of the interpolation problem. As basis functions for the interpolation space we use a parity-modified double Fourier basis on the sphere that allows us to implement the interpolation scheme in an efficient way. We further show that the numerical condition number of the interpolation scheme displays a logarithmic growth. As an application, we use the developed interpolation algorithm to estimate the rotation of an object based on measurements at the spherical Lissajous nodes.