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Advances in Mathematical PhysicsVolume 2010 2010, Article ID 457635, 37 pages

Review Article

Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

Dipartimento di Matematica ed Applicazioni, Università di Palermo, 90123 Palermo, Italy

Received 16 December 2009; Accepted 15 April 2010

Academic Editor: S. T. Ali

Copyright © 2010 Jean-Pierre Antoine and Camillo Trapani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Many families of function spaces play a central role in analysis, in particular, in signalprocessing e.g., wavelet or Gabor analysis. Typical are spaces, Besov spaces, amalgamspaces, or modulation spaces. In all these cases, the parameter indexing the family measures thebehavior regularity, decay properties of particular functions or operators. It turns out thatall these space families are, or contain, scales or lattices of Banach spaces, which are specialcases of partial inner product spaces PIP-spaces. In this context, it is often said that suchfamilies should be taken as a whole and operators, bases, and frames on them should be definedglobally, for the whole family, instead of individual spaces. In this paper, we will give an overview of PIP-spaces and operators on them, illustrating the results by space families of interest in mathematical physics and signal analysis. Theinteresting fact is that they allow a global definition of operators, and various operator classeson them have been defined.

Author: Jean-Pierre Antoine and Camillo Trapani



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