Number Theoretic Methods in Harmonic Analysis: Theory and Application

Abstract

Abstract : We have used tools from theory of harmonic analysis and number theory to extend existing theories and develop new approaches to problems. This work has focused on two areas. We have developed algorithms on extensions of Euclidean domains which have led to new computationally straightforward algorithms for parameter estimation for periodic point processes, and in particular, for sparse, noisy data. We have shown why Fourier analytic methods, e.g., Wiener's periodogram, do not produce maximum likelihood estimates for the sparse data sets on which our methods work. We are also working on extending our work to multiply periodic processes. We have also used the tools from multichannel deconvolution to develop a new procedure for multi-rate sampling We have investigated applying these techniques to develop a new procedure for A-D conversion. We have also developed interlinked wavelet bases, interlinked via number-theoretic conditions that proved useful for both multi-channel deconvolution and multi-rate sampling. We have also created new procedures for sampling in radial domains which have application to radar and sonar We have also extended these ideas to operator theory, creating sets of strongly coprime chirp and chirplet operators.