Partial inner product spaces, a unifying concept in functional analysis - Theory and applications

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, frames on them should be defined globally, for the whole family, instead of individual spaces. In this paper, we shall 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

Square integrable representations, coherent states, wavelets

We begin by quickly reviewing the basic notions of group representations, with some emphasis on unitary irreducible representations of compact groups. Then we turn to square integrable representations, the most natural generalizations of the latter. These representations are the mathematical background of the classical theory of coherent states (CS). In the next section, we examine in detail the most popular examples of this construction: (i) The continuous wavelet transform in one and two dimensions (corresponding, respectively, to the ax + b group of the line and the similitude group of the plane); and (ii) The Short-Time Fourier or Gabor transform (corresponding to the Weyl-Heisenberg group), leading to the familiar canonical coherent states. Finally, we decribe a number of recent generalizations of the classical theory, among them the so-called covariant coherent states