Stability of multiscale transformations. (English) Zbl 0919.46006
Summary: After briefly 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.
MSC:
46A35 | Summability and bases in topological vector spaces |
46B70 | Interpolation between normed linear spaces |
46M35 | Abstract interpolation of topological vector spaces |
41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
42C15 | General harmonic expansions, frames |
41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |