The work planned in this mathematical sciences group research project seeks to combine two approaches to the problem of representing signals as expansions of discrete sets of functions. One is through the Gabor transform, the other uses the wavelet expansion. One of the primary goals of this research is to combine the two approaches using a new technique called the multiplicative Zak transform. Applications of this work will concentrate on characterizations of non-stationary signal. Algorithms will be developed to compute the coefficients of the resulting double series expansion and signal properties will be derived from the expansions. Since the multiplicative Zak transformation can be interpreted as the Weil map. It appears in another context within the representation theory of the Heisenberg group. This theory provides insight into special function theory including theta functions and Krautchouk and Poisson-Charlier polynomials. Applications to number theory are also planned.
