A double Fourier sphere method for $d$-dimensional manifolds

TL;DR

This paper introduces a generalized double Fourier sphere method that transforms functions on various manifolds into a torus representation, enabling efficient Fourier approximation with uniform convergence and analytic rates.

Contribution

It extends the DFS method to a wide class of manifolds, including the rotation group, providing a unified Fourier-based approximation framework.

Findings

Uniform convergence of Fourier series on manifolds.

Analytic convergence rates for Hölder-continuous functions.

Applicable to manifolds like the sphere, disk, ball, cylinder, and rotation group.

Abstract

The double Fourier sphere (DFS) method uses a clever trick to transform a function defined on the unit sphere to the torus and subsequently approximate it by a Fourier series, which can be evaluated efficiently via fast Fourier transforms. Similar approaches have emerged for approximation problems on the disk, the ball, and the cylinder. In this paper, we introduce a generalized DFS method applicable to various manifolds, including all the above-mentioned cases and many more, such as the rotation group. This approach consists in transforming a function defined on a manifold to the torus of the same dimension. We show that the Fourier series of the transformed function can be transferred back to the manifold, where it converges uniformly to the original function. In particular, we obtain analytic convergence rates in case of H\"older-continuous functions on the manifold.