This project investigates the areas of harmonic analysis and wavelets concerning the mathematical theory of multi- dimensional wavelet expansions. One of the fundamental problems of the subject is how to construct higher dimensional wavelet bases with desired characteristics, e.g., wavelets with good time-frequency properties. These areas of research have seen significant progress due to the contributions of I. Daubechies, R. Coifman, and Y. Meyer, along with many others. However, the majority of research has been concentrated on isotropic theory, leaving many questions involving non- isotropic wavelet theory unanswered. The proposer will investigate this theory of non-isotropic wavelets from three directions. The first is to study non-isotropic analogues of the standard function spaces associated with expansive dilations. In particular, to examine characterization by wavelet expansions of Calderon-Zygmund operators associated with non-isotropic dilations. The second is to construct orthogonal wavelets with good time-frequency localization for large classes of non- isotropic expansive dilations. The third is to identify non- isotropic expansive dilations for which the construction of well-localized wavelets is impossible.
More generally, this proposal represents work on wavelet analysis which is a powerful technique in harmonic analysis. This technique has produced wide-ranging applications to signal and image processing, such as the JPEG 2000 image compression system. It is expected that further research on multi- dimensional non-isotropic wavelets will continue to produce many other contributions in pure and applied mathematics.
