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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 pseudodifferential equations on the 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.

Spline functions that interpolate data given on the sphere are developed in a weighted Sobolev space setting. The flexibility of the weights makes possible the choice of the approximating function in a way which emphasizes attributes desirable for the particular application area. Examples show that certain choices of the weight sequences yield known methods. A pointwise convergence theorem containing explicit constants yields a useable error bound.

In this paper, we deal with the problem of spherical interpolation of discretely given data of tensorial type. To this end, spherical tensor fields are investigated and a decomposition formula is described. Tensor spherical harmonics are introduced as eigenfunctions of a tensorial analogon to the Beltrami operator and discussed in detail. Based on these preliminaries, a spline interpolation process is described and error estimates are presented. Furthermore, some relations between the spline basis functions and the theory of radial basis functions are developed.