Advances in Mathematical Physics
Volume 2010 (2010), Article ID 457635, 37 pages
Review Article

The Partial Inner Product Space Method: A Quick Overview

1Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
2Dipartimento 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 signal processing (e.g., wavelet or Gabor analysis). Typical are Lp spaces, Besov spaces, amalgam spaces, or modulation spaces. In all these cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. It turns out that all these space families are, or contain, scales or lattices of Banach spaces, which are special cases of partial inner product spaces (PIP-spaces). In this context, it is often said that such families should be taken as a whole and operators, bases, and frames on them should be defined globally, 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. The interesting fact is that they allow a global definition of operators, and various operator classes on them have been defined.