Expansions in Banach Spaces

Abstract

The material presented so far 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. In this final chapter we make connections to abstract harmonic analysis and show how we can gain insight about frames via the theory for group representations. More precisely, we show how the orthogonality relations for square-integrable group representations lead to series expansions of the elements in the underlying Hilbert space; on a concrete level, this gives an alternative approach to Gabor systems and wavelet systems. Feichtinger and Gröchenig proved that the group-theoretic setup even allows us to obtain series expansions in a large scale of Banach spaces, a result which leads Gröchenig to define frames in Banach spaces. By removing some of the conditions we obtain p-frames, first studied separately by Aldroubi, Sun, and Tang.