Mathematical Sciences: Research in Fourier Analysis

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This project continues work on modern aspects of Fourier analysis related to wavelet theory, weighted inequalities for the Fourier transform and restriction theorems for the transform. Weighted Fourier transform inequalities are motivated by some of the central issues of signal processing. In linear system theory, weights correspond to various filters in energy concentration problems. Another area of application, prediction theory, weighted Lebesgue spaces arise for weights corresponding to power spectra of stationary stochastic processes. The object of the research is to obtain bounds on the q-th power norm of the weighted Fourier transform in terms of the weighted p-th power norm of the signal (original function). The program seeks best possible estimates. This search gives rise to uncertainty principle inequalities; these provide natural constraints for determining the effectiveness of spectrum estimation techniques used in signal processing. The methods of wavelet theory play a basic role in this activity. Initial efforts will focus on the special case of quadratic (weighted) norms. Inequalities may be obtained when information concerning the (integrals of) decreasing rearrangements of the weights can be computed. The same condition obtains without rearranging the weights when the inequalities are valid. What needs to be worked out is what happens between the necessary and sufficient conditions. Techniques derived from balayage combined with atomic decompositions of underlying tent spaces will be employed in efforts to clarify the different conditions.