AB-wavelet Frames in L^2(R^n)

Abstract

In order to provide a unified treatment for the
continuum and digital realm of multivariate data, Guo, Labate,
Weiss and Wilson [{\it Electron. Res. Announc. Amer. Math. Soc.}
{\bf 10} (2004), 78–-87] introduced the notion of $AB$-wavelets
in the context of multiscale analysis. We continue and extend their work
by studying the frame properties of $AB$-wavelet systems $\left\{D_{A}D_{B}T_{k}\psi^{\ell}\quad (k\in \mathbb Z^n;\;1\leqq \ell \leqq L) \right\}$ in $L^2(\mathbb R^n)$. More precisely, we establish four theorems
giving sufficient conditions under
which the $AB$-wavelet system constitutes a frame for
$L^2(\mathbb R^n)$. The proposed conditions are stated in terms of
the Fourier transforms of the generating functions.