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The Uncertainty Principle in Harmonic Analysis

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
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Book Rating : 775/5 ( reviews)

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Book Synopsis The Uncertainty Principle in Harmonic Analysis by : Victor Havin

Download or read book The Uncertainty Principle in Harmonic Analysis written by Victor Havin. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: The present book is a collection of variations on a theme which can be summed up as follows: It is impossible for a non-zero function and its Fourier transform to be simultaneously very small. In other words, the approximate equalities x :::::: y and x :::::: fj cannot hold, at the same time and with a high degree of accuracy, unless the functions x and yare identical. Any information gained about x (in the form of a good approximation y) has to be paid for by a corresponding loss of control on x, and vice versa. Such is, roughly speaking, the import of the Uncertainty Principle (or UP for short) referred to in the title ofthis book. That principle has an unmistakable kinship with its namesake in physics - Heisenberg's famous Uncertainty Principle - and may indeed be regarded as providing one of mathematical interpretations for the latter. But we mention these links with Quantum Mechanics and other connections with physics and engineering only for their inspirational value, and hasten to reassure the reader that at no point in this book will he be led beyond the world of purely mathematical facts. Actually, the portion of this world charted in our book is sufficiently vast, even though we confine ourselves to trigonometric Fourier series and integrals (so that "The U. P. in Fourier Analysis" might be a slightly more appropriate title than the one we chose).

An Introduction to the Uncertainty Principle

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 647/5 ( reviews)

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Book Synopsis An Introduction to the Uncertainty Principle by : Sundaram Thangavelu

Download or read book An Introduction to the Uncertainty Principle written by Sundaram Thangavelu. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: In 1932 Norbert Wiener gave a series of lectures on Fourier analysis at the Univer sity of Cambridge. One result of Wiener's visit to Cambridge was his well-known text The Fourier Integral and Certain of its Applications; another was a paper by G. H. Hardy in the 1933 Journalofthe London Mathematical Society. As Hardy says in the introduction to this paper, This note originates from a remark of Prof. N. Wiener, to the effect that "a f and g [= j] cannot both be very small". ... The theo pair of transforms rems which follow give the most precise interpretation possible ofWiener's remark. Hardy's own statement of his results, lightly paraphrased, is as follows, in which f is an integrable function on the real line and f is its Fourier transform: x 2 m If f and j are both 0 (Ix1e- /2) for large x and some m, then each is a finite linear combination ofHermite functions. In particular, if f and j are x2 x 2 2 2 both O(e- / ), then f = j = Ae- / , where A is a constant; and if one x 2 2 is0(e- / ), then both are null.

Toeplitz Approach to Problems of the Uncertainty Principle

2015-03-07 Mathematics
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Release : 2015-03-07
Genre : Mathematics
Kind : eBook
Book Rating : 171/5 ( reviews)

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Book Synopsis Toeplitz Approach to Problems of the Uncertainty Principle by : Alexei Poltoratski

Download or read book Toeplitz Approach to Problems of the Uncertainty Principle written by Alexei Poltoratski. This book was released on 2015-03-07. Available in PDF, EPUB and Kindle. Book excerpt: The Uncertainty Principle in Harmonic Analysis (UP) is a classical, yet rapidly developing, area of modern mathematics. Its first significant results and open problems date back to the work of Norbert Wiener, Andrei Kolmogorov, Mark Krein and Arne Beurling. At present, it encompasses a large part of mathematics, from Fourier analysis, frames and completeness problems for various systems of functions to spectral problems for differential operators and canonical systems. These notes are devoted to the so-called Toeplitz approach to UP which recently brought solutions to some of the long-standing problems posed by the classics. After a short overview of the general area of UP the discussion turns to the outline of the new approach and its results. Among those are solutions to Beurling's Gap Problem in Fourier analysis, the Type Problem on completeness of exponential systems, a problem by Pólya and Levinson on sampling sets for entire functions, Bernstein's problem on uniform polynomial approximation, problems on asymptotics of Fourier integrals and a Toeplitz version of the Beurling-Malliavin theory. One of the main goals of the book is to present new directions for future research opened by the new approach to the experts and young analysts. A co-publication of the AMS and CBMS.

Symplectic Methods in Harmonic Analysis and in Mathematical Physics

2011-07-30 Mathematics
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Release : 2011-07-30
Genre : Mathematics
Kind : eBook
Book Rating : 929/5 ( reviews)

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Book Synopsis Symplectic Methods in Harmonic Analysis and in Mathematical Physics by : Maurice A. de Gosson

Download or read book Symplectic Methods in Harmonic Analysis and in Mathematical Physics written by Maurice A. de Gosson. This book was released on 2011-07-30. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency literature. The topics that are addressed include (but are not limited to) the theory of the Wigner transform, the uncertainty principle (from the point of view of symplectic topology), Weyl calculus and its symplectic covariance, Shubin’s global theory of pseudo-differential operators, and Feichtinger’s theory of modulation spaces. Several applications to time-frequency analysis and quantum mechanics are given, many of them concurrent with ongoing research. For instance, a non-standard pseudo-differential calculus on phase space where the main role is played by “Bopp operators” (also called “Landau operators” in the literature) is introduced and studied. This calculus is closely related to both the Landau problem and to the deformation quantization theory of Flato and Sternheimer, of which it gives a simple pseudo-differential formulation where Feichtinger’s modulation spaces are key actors. This book is primarily directed towards students or researchers in harmonic analysis (in the broad sense) and towards mathematical physicists working in quantum mechanics. It can also be read with profit by researchers in time-frequency analysis, providing a valuable complement to the existing literature on the topic. A certain familiarity with Fourier analysis (in the broad sense) and introductory functional analysis (e.g. the elementary theory of distributions) is assumed. Otherwise, the book is largely self-contained and includes an extensive list of references.

Advances in Gabor Analysis

2003 Computers
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Release : 2003
Genre : Computers
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Book Rating : 396/5 ( reviews)

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Book Synopsis Advances in Gabor Analysis by : Hans G. Feichtinger

Download or read book Advances in Gabor Analysis written by Hans G. Feichtinger. This book was released on 2003. Available in PDF, EPUB and Kindle. Book excerpt: This text provides an overview of recent developments in Gabor analysis. Scientists in various disciplines related to the subject treat a range of topics from covering theory to numerics, as well as applications of Gabor analysis.

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