Stability of Multiscale Transformations

After brieey reviewing the interrelation between Riesz-bases, biorthogonality and a certain stability notion for multiscale basis transformations we establish a basic stability criterion for a general Hilbert space setting. An important tool in this context is a strengthened Cauchy inequality. It is based on direct and inverse estimates for a certain scale of spaces induced by the underlying multiresolution sequence. Furthermore, we highlight some properties of these spaces pertaining to duality, interpolation, and applications to norm equivalences for Sobolev spaces. 1 Background and Motivation A standard framework for approximately recapturing a function v in some innnite dimensional separable Hilbert space V , say, either from explicitly given data or as a solution of an operator equation, is a nested dense sequence S of closed subspaces S j where the closure is taken with respect to the norm k k V on V. Typical examples of interest are the space L 2 (() of square integrable functions on some measure space or Sobolev spaces deened over. A common feature of multiscale methods deened in such a framework, such as multigrid methods (see e.g.) is to work in one way or another with successive ne scale corrections of current approximate solutions. One possible way to formulate this is to seek for appropriate decompositions of each space S j into a direct sum of its coarser predecessor S j?1 and some complement W j