### Browsing by Author "Benedetto, John J."

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Item Fourier Transform Inequalities with Measure Weights.(1988) Benedetto, John J.; Heinig, Hans; ISRFourier transform norm inequalities, ||f^||_(q,u) <= C||f^||_(p, v'). are proved for measure weights MU on moment subspaces of L{^P AND {SUB V}}V(R^n).Density theorems are established to extend the inequalities to all of L{^P and {SUB V}}(R^n). In both cases the conditions for validity are computable. For n > 2,MU and v are radial, and the results are applied to prove spherical restriction theorems which include power weights v(t) = |t|^ALPHA,n/(p' - 1) < ALPHA < (p' + n)/(p' - 1).Item Gabor Representations and Wavelets.(1987) Benedetto, John J.; ISROur modest goal in this paper is to define a generalization of the Fourier transform of L^1(R) and to prove the L^1(R) norm inversion theorem for such a transform (Theorem 1.5). Wiener's notion of deterministic autocorrelation (from the late 1920s) arises naturally in the proof of this result; and Gabor's representation of signals (from 1946), which is a fundamental example of wavelet decomposition, provides the setting.Item A Quantitative Maximum Entropy Theorem for the Real Line.(1987) Benedetto, John J.; ISRItem Support Dependent Fourier Transform Norm Inequalities.(1986) Benedetto, John J.; Karanikas, C.; ISR(L_v)^t (R) = {f: || f ||_v = INTEGRAL: INFINITY, -INFINITY | f(t) | For a large class of weights v > O, each satisfying a natural concavity condition, the following theorems are proved for:Item THE UNCERTAINTY PRINCIPLE IN HARMONIC ANALYSIS AND BOURGAIN'S THEOREM(2003) Benedetto, John J.; Powell, Alexander M.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)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.