Lower Bounds for Finite Wavelet and Gabor Systems

Abstract

Given ψ∈L2(R) and a finite sequence {(ar,λr)}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 {(ar,λr)}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).