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Approximation properties of the double Fourier sphere method. (English) Zbl 1485.42011

Summary: 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.

MSC:

42B05 Fourier series and coefficients in several variables
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
43A90 Harmonic analysis and spherical functions
65T50 Numerical methods for discrete and fast Fourier transforms
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Software:

SHTns; NFFT
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References:

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