Decay Of The Fourier Transform
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Decay Of The Fourier Transform
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Author : Alex Iosevich
language : en
Publisher: Springer
Release Date : 2014-10-01
Decay Of The Fourier Transform written by Alex Iosevich and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-01 with Mathematics categories.
The Plancherel formula says that the L^2 norm of the function is equal to the L^2 norm of its Fourier transform. This implies that at least on average, the Fourier transform of an L^2 function decays at infinity. This book is dedicated to the study of the rate of this decay under various assumptions and circumstances, far beyond the original L^2 setting. Analytic and geometric properties of the underlying functions interact in a seamless symbiosis which underlines the wide range influences and applications of the concepts under consideration.
On The Decay Of The Fourier Transform Of Measures On Hypersurfaces Generated By Radial Functions And Related Restriction Theorems
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Author : Helmut Schulz
language : de
Publisher:
Release Date : 1990
On The Decay Of The Fourier Transform Of Measures On Hypersurfaces Generated By Radial Functions And Related Restriction Theorems written by Helmut Schulz and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with categories.
Decay Of One Dimensional Averages Of Fourier Transforms Of Measures In The Plane
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Author : Fida Michel Nashef
language : en
Publisher:
Release Date : 2011
Decay Of One Dimensional Averages Of Fourier Transforms Of Measures In The Plane written by Fida Michel Nashef and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with categories.
Consider a positive compactly supported measure in Rn, and alpha between zero and n. We are interested in obtaining decay estimates on the integral over the sphere of the Fourier transform of the above measure. Such decay estimates have important applications to the distance set problem in the area of geometric measure theory. This connection to geometric measure theory is the focus of the first two chapters of the thesis, where we also prove Sjolin's decay estimate in Rn. Our work here follows [5], [6], [7], [8], [9], and [10]. In the third chapter of the thesis, we use the Fourier restriction technology developed in in [1] and [2] to improve on Sjolin's result in dimension n=2. In the fourth chapter of the thesis, we study the analogous problem when the parabola replaces the unit circle. Our work here follows the paper [3].
Finite Fourier Transform Approximation And Riemann Sum Approximation For Functions That Decay In Time And Frequency
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Author : Jeffrey Litwin
language : en
Publisher:
Release Date : 1995
Finite Fourier Transform Approximation And Riemann Sum Approximation For Functions That Decay In Time And Frequency written by Jeffrey Litwin and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Fourier transformations categories.
Fourier Analysis And Its Applications
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Author : G. B. Folland
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
Fourier Analysis And Its Applications written by G. B. Folland and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Fourier analysis categories.
This book presents the theory and applications of Fourier series and integrals, eigenfunction expansions, and related topics, on a level suitable for advanced undergraduates. It includes material on Bessel functions, orthogonal polynomials, and Laplace transforms, and it concludes with chapters on generalized functions and Green's functions for ordinary and partial differential equations. The book deals almost exclusively with aspects of these subjects that are useful in physics and engineering, and includes a wide variety of applications. On the theoretical side, it uses ideas from modern analysis to develop the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs.
Atomic Radiative Processes
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Author : Peter R. Fontana
language : en
Publisher: Elsevier
Release Date : 2012-12-02
Atomic Radiative Processes written by Peter R. Fontana and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-02 with Science categories.
Atomic Radiative Processes provides a unified treatment of the theory of atomic radiative processes. Fourier transforms are used to obtain solutions of time-dependent Schrödinger equations, and coupled differential equations are transformed to coupled linear equations that in most cases can be readily solved. This book consists of nine chapters and begins with an overview of some of the properties of the classical field and its interaction with particles, focusing on those aspects needed for a better understanding of quantum theory. The Hamiltonian formalism is used to quantize the field, and the density of states of the radiation field is considered. The following chapters focus on a few Fourier transform techniques and their application to such areas as coherence properties of the field and amplitude and intensity correlations; the theory of angular momentum; the properties of irreducible tensors; quantization of the radiation field; and photon states. The interaction of a two-level atom with single modes of the radiation field is also discussed, along with spontaneous emission and decay processes; the evolution of coupled atomic states; the frequency distribution of emitted radiation; and radiative excitation and fluorescence. This monograph is intended for students and researchers in pure and applied physics.
Fourier Restriction For Hypersurfaces In Three Dimensions And Newton Polyhedra Am 194
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Author : Isroil A. Ikromov
language : en
Publisher: Princeton University Press
Release Date : 2016-05-24
Fourier Restriction For Hypersurfaces In Three Dimensions And Newton Polyhedra Am 194 written by Isroil A. Ikromov and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-05-24 with Mathematics categories.
This is the first book to present a complete characterization of Stein-Tomas type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface. Isroil Ikromov and Detlef Müller begin with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein-Tomas type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus Ikromov and Müller concentrate on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. They then describe decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger. Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields.
Fourier Analysis And Convexity
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Author : Luca Brandolini
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-04-27
Fourier Analysis And Convexity written by Luca Brandolini and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-04-27 with Mathematics categories.
Explores relationship between Fourier Analysis, convex geometry, and related areas; in the past, study of this relationship has led to important mathematical advances Presents new results and applications to diverse fields such as geometry, number theory, and analysis Contributors are leading experts in their respective fields Will be of interest to both pure and applied mathematicians
A Vector Field Method On The Distorted Fourier Side And Decay For Wave Equations With Potentials
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Author : Roland Donninger
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-04-26
A Vector Field Method On The Distorted Fourier Side And Decay For Wave Equations With Potentials written by Roland Donninger and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-04-26 with Mathematics categories.
The authors study the Cauchy problem for the one-dimensional wave equation ∂2tu(t,x)−∂2xu(t,x)+V(x)u(t,x)=0. The potential V is assumed to be smooth with asymptotic behavior V(x)∼−14|x|−2 as |x|→∞. They derive dispersive estimates, energy estimates, and estimates involving the scaling vector field t∂t+x∂x, where the latter are obtained by employing a vector field method on the “distorted” Fourier side. In addition, they prove local energy decay estimates. Their results have immediate applications in the context of geometric evolution problems. The theory developed in this paper is fundamental for the proof of the co-dimension 1 stability of the catenoid under the vanishing mean curvature flow in Minkowski space; see Donninger, Krieger, Szeftel, and Wong, “Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space”, preprint arXiv:1310.5606 (2013).
Fourier Hadamard And Hilbert Transforms In Chemistry
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Author : Alan Marshall
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Fourier Hadamard And Hilbert Transforms In Chemistry written by Alan Marshall and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Science categories.
In virtually all types of experiments in which a response is analyzed as a function of frequency (e. g. , a spectrum), transform techniques can significantly improve data acquisition and/or data reduct ion. Research-level nuclear magnet ic resonance and infra-red spectra are already obtained almost exclusively by Fourier transform methods, because Fourier transform NMR and IR spectrometers have been commercially available since the late 1960·s. Similar transform techniques are equally valuable (but less well-known) for a wide range of other chemical applications for which commercial instruments are only now becoming available: for example, the first corrmercial Fourier transform mass spectrometer was introduced this year (1981) by Nicolet Instrument Corporation. The purpose of this volume is to acquaint practicing chemists with the basis, advantages, and applica of Fourier, Hadamard, and Hilbert transforms in chemistry. For tions almost all chapters, the author is the investigator who was the first to apply such methods in that field. The basis and advantages of transform techniques are described in Chapter 1. Many of these aspects were understood and first applied by infrared astronomers in the 1950·s, in order to improve the otherwise unacceptably poor signal-to-noise ratio of their spec tra. However, the computations required to reduce the data were painfully slow, and required a 1 arge computer.