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Journal of Operator Theory

Volume 52, Issue 2, Fall 2004  pp. 267-291.

The local trace function of the shift invariant subspaces

Authors Dorin Ervin Dutkay
Author institution: Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, NJ 08854--8019, USA

Summary:  We define the local trace function for subspaces of $L^{2}\left(\mathbb{R}^n\right)$ which are invariant under integer translation. Our trace function contains the dimension function and the spectral function defined in \cite{BoRz} and completely characterizes the given translation invariant subspace. It has properties such as positivity, additivity, monotony and some form of continuity. It behaves nicely under dilations and modulations. We use the local trace function to deduce, using short and simple arguments, some fundamental facts about wavelets such as the characterizing equations, the equality between the dimension function and the multiplicity function and some new relations between scaling functions and wavelets.

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