The Uncertainty Principle in Harmonic Analysis and Bourgain's Theorem
The Uncertainty Principle in Harmonic Analysis and Bourgain's Theorem
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Date
2003
Authors
Powell, Alexander M.
Advisor
Benedetto, John J.
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Abstract
We investigate the uncertainty principle in harmonic analysis and how it constrains the uniform localization properties of orthonormal bases. Our main result generalizes a theorem of Bourgain to construct orthonormal bases which are uniformly well-localized in time and frequency with respect to certain generalized variances. In a related result, we calculate generalized variances of orthonormalized Gabor systems. We also answer some interesting cases of a question of H. S. Shapiro on the distribution of time and frequency means and variances for orthonormal bases.