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A unified characterization of reproducing systems generated by a finite family, II

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Abstract

By a “reproducing” method forH =L 2(ℝn) we mean the use of two countable families {e α : α ∈A}, {f α : α ∈A}, inH, so that the first “analyzes” a function h ∈H by forming the inner products {<h,e α >: α ∈A} and the second “reconstructs” h from this information:h = Σα∈A <h,e α >:f α.

A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations.

A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that “unifies” all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on ℝn. Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices.

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Authors and Affiliations

  1. Department of Mathematics, Washington University, Box 1146, 63130, St. Louis, Mo

    Eugenio Hernández, Demetrio Labate & Guido Weiss

  2. Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Eugenio Hernández

Authors
  1. Eugenio Hernández
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  2. Demetrio Labate
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  3. Guido Weiss
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Correspondence to Eugenio Hernández.

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Hernández, E., Labate, D. & Weiss, G. A unified characterization of reproducing systems generated by a finite family, II. J Geom Anal 12, 615–662 (2002). https://doi.org/10.1007/BF02930656

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  • DOI: https://doi.org/10.1007/BF02930656

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