Frames and Harmonic Analysis

The scientific objective of this proposal is the investigation of mathematical problems at the interface between frame theory and harmonic analysis. The questions and ideas have arisen during the work at the "European Center of Time-Frequency Analysis". The proposal is centered around the notion of spectral subalgebras of a Banach algebra and the applications of such algebras in frame theory and time-frequency analysis. More precisely, the proposal will pursue the following goals: a) Find and study systematic constructions of spectral subalgebras of a Banach algebra. b) Investigate algebras of infinite matrices with off-diagonal decay and the properties of their inverses. c) Investigate abstract frames over non-commutative discrete groups and extend the density theory of Balan, Casazza, Heil, and Landau to non-commutative index sets. d) Work on several open problems about Gabor frames and time-frequency analysis. What unites all four topics is the notion of spectral subalgebras. In fact, our motivation comes from time-frequency analysis where properties of spectral matrix algebras and operator algebras were used to solve concrete problems in signal analysis and wireless communications. Spectral subalgebras are so useful that we would like to investigate them as an object of independent interest. Many new questions and new ideas justify an extended research project devoted entirely to the exploration of spectral subalgebras, their usage and their applications in frame theory and in harmonic analysis. In addition the four topics stated above touch on many questions in complex analysis non- commutative harmonic analysis and operator theory and possibly also non-commutative geometry. An equally important objective of this proposal is the continuation of the "European Center of Time-Frequency Analysis" (EUCETIFA) and its maintenance in a down-sized form. This centre was founded with the support of a Marie-Curie Excellence Grant in the FP6 framework of the European Union. The Marie-Curie Excellence Grant helped to create a highly successful modern research group in time-frequency analysis (a modern branch of harmonic analysis). With team members from France, Germany, Poland, and Austria, the center had a truly European dimension and has gained high visibility and international recognition. The new project will build on the achievements and ideas of EUCETIFA and continue the work of EUCETIFA. We plan to support two postdoctoral students and one Ph.D. student for three years. This is the minimal size of a research group required to continue research on a consistently high level and maintain the international status of EUCETIFA.

This project was devoted to basic research in frame theory and harmonic analysis. Frames provide redundant representations of functions and signals.A basic question in frame theory is to quantify the redundancy of a frame. In mathematical language this is done by introducing an abstract notion of a density and then quantifying the redundancy of a frame by the density of the underlying index set, i.e., the labels of positions of the frame elements. So far a satisfactory theory for the redundancy of frames was known only for frames whose underlying index set is contained in a Euclidean space. This project developed a theory for frames over nilpotent groups. In this case the underlying index set comes with a non-Euclidean notion of distance. The main mathematical applications concern the density of sampling sets in shift-invariant spaces on nilpotent groups and frames generated by shifts and very general modulations of a single function.A particularly interesting case is the class of Gabor frames. Gabor frames are generated by shifts and modulations of a single function (a so-called window function or pulse shape), they are frequently used in speech processing and in wireless communications. Their history goes back to John von Neumann's treatment of the foundations of quantum mechanics and Dennis Gabor's development of information theory. Despite the ubiquity of Gabor frames in engineering applications there are still many difficult mathematical problems about Gabor frames. Until 2011 the complete characterization of the set of lattice parameters that generate a Gabor frame was known for exactly six (= 6) functions. An important breakthrough of this project was the design of an uncountable set of pulse shapes for which the complete set of parameters for Gabor frames can be characterized. This discovery may be relevant for many engineering applications, as it offers practitioners of time-frequency analysis a new, large class of pulse shapes with exponential decay for which the frame property is easy to determine for all lattice parameters.A third topic was the study of the quality of solutions to abstract operator equations. The problem is to understand when the solution of a linear equation in an infinite dimensional space inherits qualitative features, such as smoothness or decay, from the input function. In this project this question was treated in the context of harmonic analysis and of the construction of inverse-closed subalgebras that admit norm control.The results of this project were published in leading research journals in pure and applied mathematics.