The long road from deterministic collocation to multiresolution approximation

The aim of this report is twofold. At first, the general problem of linear approximation in Hilbert spaces is presented. The formulation and the solution of this problem is made in a very detailed way, illustrating important aspects which are often overlooked in many textbooks and monographs found in the geodetic literature. A relatively recent mathematical tool of functional analysis, called a frame, is also used to study the important issue of stability for the solution of the linear approximation problem. The second, and most important, scope of the report is to explore the relation between the choice of the Hilbert space in which we perform the linear approximation, and the stability/convergence properties of the solution for increasing data density. In particular, it will be shown that a constantly stable and convergent (in the sense of infinitely dense data) linear approximation scheme in a Hilbert space H requires the incorporation of a multiresolution structure of subspaces in H, similar to the wellknown dyadic subspace sequences in L"2(R) that were introduced by Stephane Mallat a decade ago and have created a new exciting alternative approach to wavelet theory. (orig.)SIGLEAvailable from TIB Hannover: RO 7297(1999,5) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman