Non‐orthogonal expansions on the sphere

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Abstract
Discrete families of functions with the property that every function in a certain space can be represented by its formal Fourier series expansion are developed on the sphere. A Fourier‐series‐type expansion is obviously true if the family is an orthonormal basis of a Hilbert space, but it also can hold in situations where the family is not orthogonal and is ‘overcomplete’. Furthermore, all functions in our approach are axisymmetric (depending only on the spherical distance) so that they can be used adequately in (rotation) invariant pseudo‐differential equations on the sphere. The three classes of examples particularly studied here are (i) Abel‐Poisson frames (ii) Gauss‐Weierstrass frames and (iii) frames consisting of locally supported kernel functions. Abel‐Poisson frames form families of harmonic functions and provide us with powerful approximation tools in potential theory. Gauss‐Weierstrass frames are intimately related to the diffusion equation on the sphere and play an important role in multiscale descriptions of image processing on the sphere. The third class enables us to discuss spherical ‘Fourier expansions’ by means of ‘axisymmetric finite elements’.

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