## Stability of Multiscale Transformations (1996)

Venue: | J. Fourier Anal. Appl |

Citations: | 86 - 22 self |

### BibTeX

@ARTICLE{Dahmen96stabilityof,

author = {Wolfgang Dahmen},

title = {Stability of Multiscale Transformations},

journal = {J. Fourier Anal. Appl},

year = {1996},

volume = {2},

pages = {341--361}

}

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### OpenURL

### Abstract

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. AMS Subject Classification: 41A17, 41A65, 46A35, 46B70, 46E35 Key Words: Riesz bases, biorthogonality, stability, projectors, interpolation theory, K-method, duality, Jackson, Bernstein inequalities 1 Background and Motivation A standard framework for approximately recapturing a function v in some infinite dimensional separable Hilbert space V , say, either from explicitly given data or as a solution of an operator equ...