PIP-Spaces and Signal Processing

Abstract

Contemporary signal processing makes an extensive use of function spaces, always with the aim of getting a precise control on smoothness and decay properties of functions. In this chapter, we will discuss several classes of such function spaces that have found interesting applications, namely, mixed-norm spaces, amalgam spaces, modulation spaces, or Besov spaces. It turns out that all those spaces come in families indexed by one or more parameters, that specify, for instance, the local behavior or the asymptotic properties. In general, a single space, taken alone, does not have an intrinsic meaning, it is the family as a whole that does, which brings us to the very topic of this volume. In addition, several rigged Hilbert spaces (also called Gel’fand triplets) have a particular interest, notably the one generated by the so-called Feichtinger algebra. This too deserves a detailed discussion in the sequel. Note that, unlike the previous chapter, we will treat each class with the corresponding applications. Also, we will merely state the relevant results/propositions, referring the interested reader to the vast literature quoted in the Notes.