Multi-Dimensional Wavelet and Wave Packet Expansions

Saint Louis University, Saint Louis MO

Investigators

Abstract

The PI will investigate areas of harmonic analysis concerned with multi-dimensional wavelet and wave packet expansions. In particular, he will study when collections of dilations, translations and modulations can be used to decompose square integrable functions into discrete data sets with perfect reconstruction; both with and without good time-frequency localization. This proposal relates the existence of functions useful for decomposing functions into discrete data sets to concrete geometric and algebraic questions that are of independent interest; in particular, tiling questions and the approximation of irrational numbers by rationals. Moreover, the proposer wishes to study the following phenomenon: to date, the most popular systems of the type being studied are the wavelet system (dilations and translations) and the Gabor system (dilations and modulations). Wavelets and wave packet expansions have been useful both in pure mathematics and in engineering. Wavelets are used, for example, in the FBI fingerprint compression scheme and in JPEG 2000. Wave packets have been used, for example, to give elegant proofs of the almost everywhere convergence of Fourier series. The research in this proposal will provide a better understanding of the theory behind these applications (both to math and signal processing) of wavelets and wave packets, and will potentially impact other signal processing applications and mathematics.