The Uncertainty Principle in Harmonic Analysis

1. Front Matter

   Pages I-XI

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2. Introduction

   1. Introduction

      • Victor Havin, Burglind Jöricke

      Pages 1-7

3. The Uncertainty Principle Without Complex Variables

   1. Front Matter

      Pages 9-9

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   2. Functions and Charges with Semibounded Spectra

      • Victor Havin, Burglind Jöricke

      Pages 11-52

   3. Some Topics Related to the Harmonic Analysis of Charges

      • Victor Havin, Burglind Jöricke

      Pages 53-86

   4. Hilbert Space Methods

      • Victor Havin, Burglind Jöricke

      Pages 87-116

4. Complex Methods

   1. Front Matter

      Pages 117-117

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   2. The Uncertainty Principle from the Complex Point of View. First Examples

      • Victor Havin, Burglind Jöricke

      Pages 119-195

   3. The Logarithmic Integral Diverges

      • Victor Havin, Burglind Jöricke

      Pages 197-265

   4. The Logarithmic Integral Converges

      • Victor Havin, Burglind Jöricke

      Pages 267-394

   5. Missing Frequencies and the Diameter of the Support. The Second Beurling-Malliavin Theorem and the Fabry Theorem

      • Victor Havin, Burglind Jöricke

      Pages 395-471

   6. Local and Non-local Convolution Operators

      • Victor Havin, Burglind Jöricke

      Pages 473-521

5. Back Matter

   Pages 523-547

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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).

• Fourier transform
• Fouriertransformation
• Newton Potential
• Potential
• Quasi-Analysierbarkeit
• Quasianalytizität
• Uncertainty Principle
• fourier analysis
• functional analysis
• harmonic analysis

• Department of Mathematics, St. Petersburg State University, St. Petersburg, Russia

  Victor Havin

• Max-Planck-Gesellschaft zur Förderung der Wissenschaften, Berlin, Germany

  Burglind Jöricke

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