In harmonic analysis, frames provide decompositions of functions where the individual components (of the frame) are not necessarily linearly independent. In recent years, they have become the standard tool for generating useful redundant (or over complete) signal representations. Frames make signal representations more flexible, robust to transmission errors, and resilient to noise. Despite the prevalence of stochastic models for noise and other uncertainties that plague many signal processing problems, deterministic perspectives and methods remain dominant in frame theory research.
The pervasive theme of this research is to study how well (stochastic) signals can be represented by random subframes of standard ones. The ultimate goal is to design/study frames and random sub-frames that are resilient to transmission losses or erasures of frame coefficients. This will entail developing notions of optimality under erasure and characterizations of optimal frames. The application of this theoretical work to signal processing problems will promote the transition of basic mathematical results into applications and perhaps ultimately into new technologies. Since the proposed study lies in the interface of many mathematical sub-disciplines including harmonic analysis, and probability theory, it will also enable dissemination of mathematical knowledge across sub-disciplines and will foster an instructional agenda that embraces this cross-disciplinary view. Elements of the proposed research will be incorporated into a newly developed course (by PI) in approximation theory at the University of Idaho, as well as, student research activities.
