# Browsing by Subject "Harmonic analysis"

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Item Combinatorial and probabilistic techniques in harmonic analysis(2012-05) Lewko, Mark J., 1983-; Vaaler, Jeffrey D.; Beckner, William; Pavlovic, Natasa; Rodriguez-Villegas, Fernando; Zuckerman, DavidShow more We prove several theorems in the intersection of harmonic analysis, combinatorics, probability and number theory. In the second section we use combinatorial methods to construct various sets with pathological combinatorial properties. In particular, we answer a question of P. Erdos and V. Sos regarding unions of Sidon sets. In the third section we use incidence bounds and bilinear methods to prove several new endpoint restriction estimates for the Paraboloid over finite fields. In the fourth and fifth sections we study a variational maximal operators associated to orthonormal systems. Here we use probabilistic techniques to construct well-behaved rearrangements and base changes. In the sixth section we apply our variational estimates to a problem in sieve theory. In the seventh section, motivated by applications to sieve theory, we disprove a maximal inequality related to multiplicative characters.Show more Item Coordinate-free principles for extension of smooth functions(2021-06-23) Frei-Pearson, Abraham; Israel, Arie, 1988-; Beckner, William; Fefferman, Charles; Maggi, FrancescoShow more This dissertation will study interpolation of smooth functions, broadly defined, in two related contexts. Given a finite subset E of ℝ [superscript n] and a function f : E→ℝ , what is the smallest C [superscript m]-norm of a function F : ℝ [superscript n]→ℝ extending f? In chapter 2, we prove the following result: for every m,n, there exist constants k [superscript #] and C [superscript #] depending on m and n only such that the following holds. Suppose that for every set S ⊂ E with at most k [superscript #] points, there exists a function F [superscript S] : ℝ [superscript n] → ℝ such that F [superscript S] |[subscript S] = f|[subscript S], and [double bar]F [superscript S] [double bar] [subscript C superscript m][subscript parenthesis ℝ superscript n parenthesis] ≤ 1. Then there exists a function F extending f of C [superscript m]-norm at most C [superscript #]. Our approach to this theorem is coordinate-free, and establishes constants which are an exponential improvement over the constants previously established in the literature. Our results are proved by induction on a measure of the tameness of the set E. In order to control the number of steps in the induction argument, we must identify a certain quantity called the signature which has crucial monotonicity properties. In chapter 3, we investigate a related problem. Suppose (X, d) is a metric space, and Γ is a map from X into the compact, convex subsets of the hyperbolic plane ℍ². We are interested in constructing a Lipschitz selection of Γ, i.e. a Lipschitz map F : X → ℍ² such that F(x) ∈ Γ (x) for all x ∈ X. We establish the following results: there exist universal constants k [superscript #] and C [superscript #] independent of Γ and X such that the following holds. Suppose that, for all subsets S ⊂ X containing at most k [superscript #] points, there is a map F [superscript S] : S → ℍ² of Lipschitz constant 1 such that F [superscript S] (x) ∈ Γ (x) for all x ∈ S. Then there is a map F : X → ℍ² such that F(x) ∈ Γ(x) for all x ∈ X with Lipschitz constant at most C [superscript #].Show more Item Extremal problems in Fourier analysis, Whitney's theorem, and the interpolation of data(2019-08) Carruth, Jacob Thomas; Israel, Arie, 1988-; Beckner, William; Ward, Rachel; Carneiro, EmanuelShow more This dissertation deals with three problems in interpolation theory. The first two, the Beurling-Selberg box minorant problem and Turán's extremal problem, are optimization problems involving constrained interpolation by bandlimited functions. The Beurling-Selberg box minorant problem is a higher dimensional version of Selberg's minorant problem for the interval. We study the problem of minorizing the indicator function of the unit cube Q [subscript d] = [-1, 1] [superscript d] by a function bandlimited to Q [subscript d]. We show that there exists a dimension d* ≤ 710 such that if d > d* then there do not exist d-dimensional minorants. We also construct the first non-trivial minorants for dimensions 2, 3, 4, and 5. Next, we show how to compute upper and lower bounds for the value of Turán's extremal problem by solving finite dimensional linear programs. The problem depends on a convex body K; our bounds have been used to compute the sharpest known upper bound in the case in which K is the 3 dimensional ℓ₁ ball. The third problem we study concerns the interpolation of data by C [superscript m] functions. We give a new proof of the Brudnyi-Shvartsman-Fefferman finiteness principle for C [superscript m-1,1] (R [superscript d]) functions. We hope that this proof will lead to practical algorithms for C[superscript m] interpolationShow more Item Extremality, symmetry and regularity issues in harmonic analysis(2009-05) Carneiro, Emanuel Augusto de Souza; Beckner, WilliamShow more In this Ph. D. thesis we discuss four different problems in analysis: (a) sharp inequalities related to the restriction phenomena for the Fourier transform, with emphasis on some Strichartz-type estimates; (b) extremal approximations of exponential type for the Gaussian and for a class of even functions, with applications to analytic number theory; (c) radial symmetrization approach to convolution-like inequalities for the Boltzmann collision operator; (d) regularity of maximal operators with respect to weak derivatives and weak continuity.Show more Item Flexible fitting in 3D EM(2012-12) Bettadapura Raghu, Prasad Radhakrishna; Bajaj, Chandrajit; Crawford, Richard; Chen, Dongmei; Truskett, Thomas; Ying, LexingShow more In flexible fitting, the high-resolution crystal structure of a molecule is deformed to optimize its position with respect to a low-resolution density map. Solving the flexible fitting problem entails answering the following questions: (A) How can the crystal structure be deformed? (B) How can the term "optimum" be defined? and (C) How can the optimization problem be solved? In this dissertation, we answer the above questions in reverse order. (C) We develop PFCorr, a non-uniform SO(3)-Fourier-based tool to efficiently conduct rigid-body correlations over arbitrary subsets of the space of rigid-body motions. (B) We develop PF2Fit, a rigid-body fitting tool that provides several useful definitions of the optimal fit between the crystal structure and the density map while using PFCorr to search over the space of rigid-body motions (A) We develop PF3Fit, a flexible fitting tool that deforms the crystal structure with a hierarchical domain-based flexibility model while using PF2Fit to optimize the fit with the density map. Our contributions help us solve the rigid-body and flexible fitting problems in unique and advantageous ways. They also allow us to develop a generalized framework that extends, breadth-wise, to other problems in computational structural biology, including rigid-body and flexible docking, and depth-wise, to the question of interpreting the motions inherent to the crystal structure. Publicly-available implementations of each of the above tools additionally provide a window into the technically diverse fields of applied mathematics, structural biology, and 3D image processing, fields that we attempt, in this dissertation, to span.Show more Item Methods of dynamical systems, harmonic analysis and wavelets applied to several physical systems(2002) Petrov, Nikola Petrov; Llave, Rafael de laShow more Item Nine-Phase Armature Windings Design, Test, and Harmonic Analysis(IEEE, 2005-05) Jordan, H.E; Zowarka, R.C.; Pratap, S.B.Show more A nine-phase armature winding was developed for a large generator. Alternative methods for interconnecting the pole-phase groups were examined. An alternate-pole connection scheme was adopted and a prototype induction motor was constructed to confirm the winding scheme. Since only a three-phase power source was available for testing, the induction motor was tested by using three, three-phase winding sections, one at a time. Air-gap harmonic fields produced some unusual results. These test results and harmonic analyses to explain them are presented herein. The tests confirmed the nine-phase winding scheme that was adopted. The harmonic analyses revealed that the complete nine-phase winding exhibited a very low harmonic content, a distinct advantage of a nine-phase winding for future applications.Show more