Irregular wavelet frames and Gabor frames

Given g∈L2(Rn), we consider irregular wavelet for the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j $$ \end{document} > 0 and b > 0.

Sufficient conditions for the wavelet system to constitute a frame for L2(Rn) are given. For a class of functions g∈L22(Rn) we prove that certain growth conditions on {λj} will frames, and that some other types of sequences exclude the frame property. We also give a sufficient condition for a Gabor system\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} $$ \end{document}to be a frame.