Four main directions are selected for advancement in this project: (a) Time-frequency independence conjectures and related topics; (b) Quantification of localization for nonabelian coherent state frames; (c) Applications to affine pseudodifferential operators (broadband time-varying filters); (d) Multiwavelet development of the shearlet transform and construction of compactly supported shearlets. Each of these interrelating topics contains deep mathematical questions directly related to fundamental issues in redundant representation and transformation of signals. Each topic reaches into a broad range of areas in mathematical analysis, and progress requires a synthesis of different ideas from both abstract analysis and applied harmonic analysis, including pseudodifferential operator theory, noncommutative Banach algebras, and new forms of multiresolution analysis suitable for two-dimensional and higher signal processing.
The problems addressed in this project concern fundamental questions regarding the basic mathematical tools that underlie modern signal processing, including such applications as wireless communications, radar, seismic sensing, and image processing. Progress on these topics yields applicable mathematics with potential impact for real-world scientific and engineering programs. All aspects of this work are intimately tied with educational endeavors, both at the graduate and undergraduate levels. Students of the investigator cooperate directly in this research, receiving a breadth and depth of knowledge and experience that will foster a continued commitment to cross-disciplinary interactions among mathematics, physics, and engineering. Topics related to this project and progress on the problems of the project are incorporated into courses and seminars directed by the investigator. The investigator continues his strong record of producing graduate students, working with undergraduates, mentoring postdocs, producing Ph.D. students from underrepresented groups, and encouraging underrepresented groups to pursue research.
