Irregular wavelet frames and gabor frames

Abstract

Given g ∈ L^2(R^n), we consider irregular wavelet systems of the form {λ^{n/2}_j g(λ_jx − kb)}j∈Z,k∈Z^n , where λ_j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L^2(R^n) are given. For a class of functions g ∈ L^2 (R^n) we prove that certain growth conditions on {λ_j} will lead to frames, and that some other types of sequences exclude the frame property. We also give a sufficient condition for a Gabor system {e^{2πib(j,x)}g(x − λ_k}j∈Z^n, k∈Z to be a frame.