Abstract:

We give an explicit construction for a system of interpolation
nodes and the corresponding interpolation basis for a space that
allows a discrete fast Fourier transform type matrix-free formula
for interpolating functions on the sphere. We prove that the
quality of the spherical interpolation operator is the same as that
of the classical spectral interpolation operator for two dimensional
periodic functions. We also construct a minimal quadrature rule
for the space (with number of points equal to the dimension of the
space), and describe an associated interpolation operator.
Finally, given a large dataset in the latitudinal and longitudinal
directions, with a longitudinal symmetry condition, we construct a
quasi-interpolatory approximation of the function representing the
scattered data. The quasi-interpolatory operator gives near best
approximation to every continuous function on the sphere. We
demonstrated our matrix-free operators with several benchmark
numerical experiments.