Approximation Properties of the Double Fourier Sphere Method

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 Hölder 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.

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

“…Thereby, a function f : S 2 → C is concatenated with the DFS coordinate transform φ S 2 : T 2 → S 2 , (x 1 , x 2 ) → (cos x 1 sin x 2 , sin x 1 sin x 2 , cos x 2 ), which covers the sphere twice. The transform φ S 2 is smooth, and the transformed function f • φ S 2 has a convergent Fourier series for sufficiently smooth f , see [31]. Furthermore, we have φ S 2 (x 1 , x 2 ) = φ S 2 (x 1 + π, −x 2 ) for (x 1 , x 2 ) ∈ T 2 , so that the transformed function is block-mirror-centrosymmetric (BMC), cf.…”

“…The classical DFS method was originated in 1972 by Merilees [29] and found various applications since, e.g., [4,8,13,16,34,36,39,46,48,49]. Recently, we have shown analytic approximation properties of the classical DFS method [31]. Further DFS methods have been invented for other geometries such as the disk [47], the cylinder [18], the ball [3], and two-dimensional axisymmetric surfaces [32].…”

“…We define a generalization of the classical DFS method to other manifolds in a way that covers generalizations from the literature, such as ball and cylinder, and yields results analogous to those we presented in [31] for the sphere.…”

“…Because the restriction of φ S 2 to (−π, π] × (0, π) ∪ {(0, 0), (0, π)} is bijective, there is a one-to-one connection between BMC-functions on the torus and functions on the sphere without its poles. This makes it possible to relate the Fourier expansion of a transformed function f • φ S 2 to a series expansion of f defined directly on the sphere, see [31].…”

A Frame Decomposition of the Funk-Radon Transform

“…In particular, we construct an FD using trigonometric functions, which have the advantage of their fast computation outperforming spherical harmonics, cf. [35,53]. For this, we first review some background on frames and FDs in Section 2.…”