Wavelet frames with irregular matrix dilations and their stability ✩

Let ψ ∈ L2(Rn), B and Aj (j ∈ Z) be real nonsingular n × n matrices, λk (k ∈ Zn) be real numbers. In this paper we present a sufficient condition for the system {|detAj |1/2ψ(Aj x − Bλk): j ∈ Z, k ∈ Zn} to be a frame for L2(Rn). This sufficient condition also shows the stability of the system with respect to the perturbation of matrix dilation parameters {Aj }j∈Z and the perturbation of translation parameters {λk}k∈Zn .  2004 Elsevier Inc. All rights reserved