The Characterization of Parseval Frame Wavelets

Abstract:

      In this paper, we characterize all generalized low pass filters and MRA Parseval frame wavelets in $L^2(R^n)$ with matrix dilations of the form $(D f )(x) = \sqrt{2}f (Ax)$, where $A$ is an arbitrary expanding $n\times n$ matrix with integer coefficients, such that $|{\rm det}\,A| =2$. We study the pseudo-scaling functions, generalized low pass filters and MRA Parseval frame wavelets and give some important characterizations about them. Furthermore, we give a characterization of the semiorthogonal MRA Parseval frame wavelets and provide several examples to verify our results.