## 65D15 Algorithms for functional approximation

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The thesis is concerned with multiscale approximation by means of radial basis functions on hierarchically structured spherical grids. A new approach is proposed to construct a biorthogonal system of locally supported zonal functions. By use of this biorthogonal system of locally supported zonal functions, a spherical fast wavelet transform (SFWT) is established. Finally, based on the wavelet analysis, geophysically and geodetically relevant problems involving rotation-invariant pseudodifferential operators are shown to be efficiently and economically solvable.

We introduce a method to construct approximate identities on the 2-sphere which have an optimal localization. This approach can be used to accelerate the calculations of approximations on the 2-sphere essentially with a comparably small increase of the error. The localization measure in the optimization problem includes a weight function which can be chosen under some constraints. For each choice of weight function existence and uniqueness of the optimal kernel are proved as well as the generation of an approximate identity in the bandlimited case. Moreover, the optimally localizing approximate identity for a certain weight function is calculated and numerically tested.

Wavelet transform originated in 1980's for the analysis of seismic signals has seen an explosion of applications in geophysics. However, almost all of the material is based on wavelets over Euclidean spaces. This paper deals with the generalization of the theory and algorithmic aspects of wavelets to a spherical earth's model and geophysically relevant vector fields such as the gravitational, magnetic, elastic field of the earth.A scale discrete wavelet approach is considered on the sphere thereby avoiding any type of tensor-valued 'basis (kernel) function'. The generators of the vector wavelets used for the fast evaluation are assumed to have compact supports. Thus the scale and detail spaces are finite-dimensional. As an important consequence, detail information of the vector field under consideration can be obtained only by a finite number of wavelet coefficients for each scale. Using integration formulas that are exact up to a prescribed polynomial degree, wavelet decomposition and reconstruction are investigated for bandlimited vector fields. A pyramid scheme for the recursive computation of the wavelet coefficients from level to level is described in detail. Finally, data compression is discussed for the EGM96 model of the earth's gravitational field.