Harmonic and Functional Analysis of Wavelet and Frame Expansions

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The project involves research and education activities in harmonic and functional analysis concerning the mathematical theory of multidimensional wavelet and frame expansions. Wavelet and frame theory is not only mathematically interesting in its own right, but this area has found many applications outside of pure mathematics. These range from applied and computational harmonic analysis to signal processing and data compression. Some well known examples where wavelets are a key tool include the JPEG 2000 digital image standard and fingerprint compression for data storage. The broader impacts of the project deal with education and training of undergraduate and graduate students in the area of harmonic analysis and wavelets. There are three components of this activity which are integrally connected with the research aims of this project. The first is advising and training of Ph.D. students in the research directions described in this project. The second educational goal is a support for student-run Stochastic/Functional Analysis seminar, which will provide additional, critical training for graduate students interested in doing research in harmonic and functional analysis. The third aim is to recruit advanced undergraduate students to research projects on frames. The project aims at answering some of the most fundamental questions in wavelet and frame theory. One of the main research directions of the project is the development of techniques for construction of well-localized orthogonal wavelets for large classes of nonisotropic expanding dilations. A closely related complementary topic is the study of wavelets for nonexpanding dilations. The problem of characterizing dilations for which there exist minimally supported frequency (MSF) wavelets is connected with the geometry of numbers; more specifically, with the estimate on the number of lattice points of dilates of balls. An analogous problem of classifying dilations for which there exist well-localized wavelets has implications in the study of nonisotropic function spaces. Another direction of the project is the construction of frames with desired properties, such as with prescribed norms and frame operator. This line of research is closely related with the infinite-dimensional generalizations of the Schur-Horn theorem. The problem of characterizing diagonals of self-adjoint operators has not only implications for frame theory, but it has also been extensively studied in the setting of von Neumann algebras. Finally, the problem of characterizing tight fusion frame sequences is connected with algebraic combinatorics methods that are typically not employed in analysis.