Lower bounds for finite wavelet and Gabor systems (Q2744429)

In this paper the authors obtain explicit lower frame bounds for some finite wavelet and Gabor systems. For a finite family \(\{ f_{\gamma} {\}}_{\gamma\in\Gamma}\) of vectors of a Hilbert space \(H\), a lower frame bound \(A\) is any positive number \(A>0\) such that \(\sum_{\gamma}|\langle f,f_{\gamma}\rangle|^2 \geq A \|f \|^2\), for all \(f\in \text{span}\{ f_{\gamma} \}\). Given \(\Psi\in L^2(R)\) and a finite sequence \(\{(a_{\gamma},\lambda_{\gamma}) {\}}_{\gamma\in\Gamma} \subset {R}^{+}\times R\) consisting of distinct points, the corresponding wavelet system is the set of functions \(\{ {1 \over {a_{\gamma}^{1/2}}} \Psi( { x \over {a_{\gamma}}}- {\lambda}_{\gamma}) {\}}_{\gamma\in\Gamma}\). For \(g\in L^2(R)\) and finite \(\{(a_{\gamma},\lambda_{\gamma}) {\}}_{\gamma\in\Gamma}\subset R\times R\), the Gabor system is defined by \(\{ e^{2\pi ia_{\gamma}x}g(x-{\lambda}_{\gamma}) {\}}_{\gamma\in\Gamma}\). The authors prove that for a dense set of functions \(\Psi\) (or \(g\)) in \(L^2(R)\), the wavelet (resp. Gabor) system corresponding to any choice of \(\{(a_{\gamma},\lambda_{\gamma}) {\}}_{\gamma\in\Gamma}\) is linearly independent, and explicit estimates of the lower frame bounds are derived.