Kaiserslautern - Fachbereich Mathematik
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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.
This paper presents a wavelet analysis of temporal and spatial variations of the Earth's gravitational potential based on tensor product wavelets. The time--space wavelet concept is realized by combining Legendre wavelets for the time domain and spherical wavelets for the space domain. In consequence, a multiresolution analysis for both, temporal and spatial resolution, is formulated within a unified concept. The method is then numerically realized by using first synthetically generated data and, finally, several real data sets.
A new class of locally supported radial basis functions on the (unit) sphere is introduced by forming an infinite number of convolutions of ''isotropic finite elements''. The resulting up functions show useful properties: They are locally supported and are infinitely often differentiable. The main properties of these kernels are studied in detail. In particular, the development of a multiresolution analysis within the reference space of square--integrable functions over the sphere is given. Altogether, the paper presents a mathematically significant and numerically efficient introduction to multiscale approximation by locally supported radial basis functions on the sphere.