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Approximation Theory and Its Applications

, Volume 17, Issue 1, pp 18–29 | Cite as

Lower Bounds for Finite Wavelet and Gabor Systems

  • Ole Christensen
  • Alexander M. Lindner
Article
  • 39 Downloads

Abstract

Given ψ∈L2(R) and a finite sequence {(a r r )}r∈Γ⫅R+XR consisting of distinct points, the corresponding wavelet system is the set of functions\(\left\{ {\frac{1}{{a_\gamma ^{1/2} }}\phi (\frac{x}{{a_\gamma }} - \lambda _\gamma )\gamma \varepsilon r} \right\}\). We prove that for a dense set of functions ψ∈L2(R) the wavelet system corresponding to any choice of {(a r r )}r∈Γis linearly independent, and we derive explicite estimates for the corresponding lower (frame) bounds. In particular, this puts restrictions on the choice of a scaling function in the theory for multiresolution analysis. We also obtain estimates for the lower bound for Gabor systems\(\left\{ {e^{2rie_{\gamma x} } g(x - \lambda _\gamma )} \right\}\gamma \varepsilon r\)for functions g in a dense subset of L2(R).

Keywords

Lower Bound Distinct Point Dense Subset Scaling Function Finite Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Ole Christensen
    • 1
  • Alexander M. Lindner
    • 2
  1. 1.Department of MathematicsTechnical University of DenmarkLyngbyDenmark
  2. 2.Department of MathematicsTechnical University of MunichMunichGermany

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