Expansions in Banach Spaces

Abstract

The material presented in this book naturally splits in two parts: a functional analytic treatment of frames in general Hilbert spaces, and a more direct approach to structured frames like Gabor frames and wavelet frames. For the second part the most general results were presented in Chapter 21, in the setting of generalized shift-invariant systems on an LCA group.The current chapter is in a certain sense a natural continuation of both tracks. We consider connections between frame theory and abstract harmonic analysis and show how we can construct frames in Hilbert spaces via the theory for group representations. In special cases the general approach will bring us back to the Gabor systems and wavelet systems. The abstract framework adds another new aspect to the theory: we will not only obtain expansions in Hilbert spaces but also in a class of Banach spaces.