Orthogonal Zonal, Tesseral and Sectorial Wavelets on the Sphere for the Analysis of Satellite Data

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Abstract

The spherical harmonics [Yn, k]n = 0,1,…; k = −n,…, n represent a standard complete orthonormal system in ℒ2(Ω), where Ω is the unit sphere. In view of present and future satellite missions (e.g., for the determination of the Earth's gravity field) it is of particular importance to treat the different accuracies and sizes of data in dependence of the index pairs (n, k). It is, e.g., known that the GOCE mission yields essentially less accurate data in the zonal (k=0) case. Therefore, this paper presents new ways of constructing multiresolutions for a Sobolev space of functions on Ω allowing the separate treatment of certain classes of pairs (n, k) and, in particular, the separate treatment of different orders k. Orthogonal bandlimited as well as non-bandlimited detail and scale spaces adapted to certain (geo)scientific problems and to the character of the given data can now be used. Finally, an explicit representation of a non-bandlimited wavelet on Ω yielding an orthogonal decomposition of the function space is calculated for the first time.

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