Lower Bounds for Finite Wavelet and Gabor Systems

Abstract

Given ψ∈L ²(R) and a finite sequence {(a _r ,λ _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∈L ²(R) the wavelet system corresponding to any choice of {(a _r ,λ _r )} _r∈Γ is 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 L ²(R).