The Partial Inner Product Space Method: A Quick Overview

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