Harmonic and functional analysis of wavelet and frame expansions

The project involves research and education activities in harmonic and functional analysis concerning the mathematical theory of multi-dimensional wavelet and frame expansions. Wavelet and frame theory is not only mathematically interesting as a subject of the study by itself, but this area has found many applications outside of pure mathematics ranging 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 the education and training of undergraduate and graduate students in the area of harmonic analysis and wavelets. The project aims to answer 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 the construction of well-localized orthogonal wavelets for large classes of non-isotropic expanding dilations. A closely related complementary topic is the study of wavelets for non-expanding dilations. A recent solution of the wavelet set problem by the PI and Speegle, 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. 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 to 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 PI aims to investigate the Akemann-Weaver conjecture, which is a higher-rank extension of Weaver’s conjecture that was proven by Marcus, Spielman, and Srivastava in their breakthrough solution of the Kadison-Singer problem. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.