The Partial Inner Product Space Method: A Quick Overview
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,...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2010/457635 |
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| author | Jean-Pierre Antoine Camillo Trapani |
| author_facet | Jean-Pierre Antoine Camillo Trapani |
| author_sort | Jean-Pierre Antoine |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-7425b065222f4387b60a30234840a242 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-7425b065222f4387b60a30234840a2422025-02-03T01:26:51ZengWileyAdvances in Mathematical Physics1687-91201687-91392010-01-01201010.1155/2010/457635457635The Partial Inner Product Space Method: A Quick OverviewJean-Pierre Antoine0Camillo Trapani1Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, 1348 Louvain-la-Neuve, BelgiumDipartimento di Matematica ed Applicazioni, Università di Palermo, 90123 Palermo, ItalyMany 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.http://dx.doi.org/10.1155/2010/457635 |
| spellingShingle | Jean-Pierre Antoine Camillo Trapani The Partial Inner Product Space Method: A Quick Overview Advances in Mathematical Physics |
| title | The Partial Inner Product Space Method: A Quick Overview |
| title_full | The Partial Inner Product Space Method: A Quick Overview |
| title_fullStr | The Partial Inner Product Space Method: A Quick Overview |
| title_full_unstemmed | The Partial Inner Product Space Method: A Quick Overview |
| title_short | The Partial Inner Product Space Method: A Quick Overview |
| title_sort | partial inner product space method a quick overview |
| url | http://dx.doi.org/10.1155/2010/457635 |
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