Approximation properties of the double Fourier sphere method

TL;DR

This paper analyzes the approximation capabilities of the double Fourier sphere method, showing how it preserves smoothness and converges under certain conditions when transforming functions from the sphere to the torus.

Contribution

It provides a detailed analysis of the DFS method's analytic properties, including smoothness preservation and convergence criteria, which were previously not fully understood.

Findings

DFS preserves smoothness in Hölder spaces

DFS does not preserve Sobolev spaces in the same way

Conditions for absolute convergence and convergence speed are established

Abstract

We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written as a Fourier series yielding an approximation of the original function. We show that the DFS method preserves smoothness: it continuously maps spherical H\"older spaces into the respective spaces on the torus, but it does not preserve spherical Sobolev spaces in the same manner. Furthermore, we prove sufficient conditions for the absolute convergence of the resulting series expansion on the sphere as well as results on the speed of convergence.