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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1687-9120 1687-9139 |