Lower bounds for finite wavelet and Gabor systems

Given ψ ∈ L2(R) and a finite sequence {(aγ , λγ)}γ∈Γ ⊆ R+ × R consisting of distinct points, the corresponding wavelet system is the set of functions { 1 a 1/2 γ ψ( x aγ − λγ)}γ∈Γ. We prove that for a dense set of functions ψ ∈ L2(R), the wavelet system corresponding to any choice of {(aγ , λγ)}γ∈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 {e2πiaγxg(x − λγ)}γ∈Γ for functions g in a dense subset of L2(R).