NECESSARY CONDITION AND SUFFICIENT CONDITION FOR THE WAVELET FRAMES IN

The main goal for this paper is consider the necessary conditions and sufficient conditions of wavelet frames in higher dimensions with an arbitrary expanding matrix dilation. At first, we give a necessary condition of wavelet frame in $L^2(R^n)$, which generalizes the univariate results of Shi from one dimension with an arbitrary real number a(a > 1) dilation to higher dimension with an arbitrary expansive matrix dilation. Secondly, we deduce a necessary condition for wavelet frames in $L^2(R^n)$ when the function $\psi$ satisfies some property of the decay. For the case n = 1, we obtain a corollary which has weaker condition comparing with existing result.