Orthogonal zonal, tesseral and sectorial wavelets on the sphere for the analysis of satellite data

The spherical harmonics {Yn, k}n= 0,1,...; k=- n,..., n represent a standard complete orthonormal system in L-2(Omega), where Omega 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 Omega 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 Omega yielding an orthogonal decomposition of the function space is calculated for the first time.