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Time-shifts Generalized Multiresolution Analysis over Dyadic-Scaling Reducing Subspaces


\begin{abstract} A Generalized Multiresolution Analysis (GMRA) associated with a wavelet is a sequence of nested subspaces of the function space $\L^2(\RR)$, with specific properties, and arranged in such a way that each of the subspaces corresponds to a scale $2^m$ over {\it all}\/ time-shifts $n$. These subspaces can be expressed in terms of a generating-wandering subspace --- of the dyadic-scaling operator --- spanned by orthonormal wavelet-functions --- generated from the wavelet. In this article we show that a GMRA can also be expressed in terms of subspaces for each time-shift $n$ over all scales $2^m$. This is achieved by means of ``elementary'' reducing subspaces of the dyadic-scaling operator. Consequently, Time-Shifts GMRA associated with wavelets, as well as ``sub-GMRA'' associated with ``sub-wavelets'' will then be introduced. \end{abstract}

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