Operator Representations of Frames and Structured Function Systems

The thesis deals with frames in infinite-dimensional Hilbert spaces. The key aspects are to analyze frames that can be represented via iterates of a bounded operator and to develop the theory for generalized shift-invariant frames. The generalized shift-invariant systems provide a general framework to analyze large classes of structured function systems, e.g., shift-invariant systems, Gabor systems, wavelet systems, wave packet systems, etc.

The thesis consists of five published research papers, three of which devoted to operator representations of frames. The desire of representing a frame as iterated actions of a bounded operator is motivated by the recent research topic “dynamical sampling” and has roots in as well linear dynamics as applied harmonic analysis. We will consider two versions of iterated systems, namely, systems indexed by N∪{0} respectively Z. The indexing by N∪{0} comes up naturally in dynamical sampling, and the indexing by Z is motivated by several classical examples in applied harmonic analysis. A further motivation is that several natural structured function systems in applications, e.g., Gabor systems and wavelet systems, are countable unions of iterated systems. One of the key contributions consists in a detailed analysis of frames for which an operator representation is available, in particular, with a bounded operator.

The last two papers deal with various aspects of structured function systems. Motivated by the fact that linear independency turned out to be one of the key conditions in the analysis of iterated systems, we consider discrete Gabor systems generated by a finite sequence and characterize linear independency in terms of the involved parameters and the support size. Finally, we consider generalized shift-invariant systems and derive a condition that is necessary and sufficient for the socalled Calderón sum to be bounded below. We also provide a construction of dual pairs of generalized shift-invariant frames. The construction generalizes the known results for Gabor frames and wavelet frames and also apply to other classes of frames, e.g., wave packet frames.