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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
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