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Commutative Harmonic Analysis III

Generalized Functions. Application

  • Book
  • © 1995

Overview

Editors:
  1. V. P. Havin

    1. Department of Mathematics, St. Petersburg State University, St. Petersburg, Staryj Peterhof, Russia
      Department of Mathematics and Statistics, Mc Gill University, Montreal, Canada

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  2. N. K. Nikol’skij

    1. Steklov Mathematical Institute, St. Petersburg, Russia
      Département de Mathématiques, Université de Bordeaux I, Talence, Cedex, France

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    You can also search for this editor in PubMed   Google Scholar

Part of the book series: Encyclopaedia of Mathematical Sciences (EMS, volume 72)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-vii
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  2. Distributions and Harmonic Analysis

    1. Front Matter

      Pages 1-2
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    2. Introduction

      • V. P. Havin, N. K. Nikol’skij
      Pages 3-4

    3. The Elementary Theory

      • V. P. Havin, N. K. Nikol’skij
      Pages 5-14

    4. The General Theory

      • V. P. Havin, N. K. Nikol’skij
      Pages 15-38

    5. The Fourier Transform

      • V. P. Havin, N. K. Nikol’skij
      Pages 38-59

    6. Special Problems

      • V. P. Havin, N. K. Nikol’skij
      Pages 60-83

    7. Contact Structures and Distributions

      • V. P. Havin, N. K. Nikol’skij
      Pages 83-120

  3. Optical and Acoustic Fourier Processors

    1. Front Matter

      Pages 129-130
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    2. Introduction

      • V. P. Havin, N. K. Nikol’skij
      Pages 131-135

    3. The Optical Fourier Transform

      • V. P. Havin, N. K. Nikol’skij
      Pages 136-147

    4. Notes and Comments

      • V. P. Havin, N. K. Nikol’skij
      Pages 147-159

    5. Acoustic and Acousto-Optical Fourier Processors

      • V. P. Havin, N. K. Nikol’skij
      Pages 160-173

  4. The Uncertainty Principle in Harmonic Analysis

    1. Front Matter

      Pages 177-179
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    2. Introduction

      • V. P. Havin, N. K. Nikol’skij
      Pages 180-180

    3. The Uncertainty Principle Without Complex Variables

      • V. P. Havin, N. K. Nikol’skij
      Pages 181-207

    4. Complex Methods

      • V. P. Havin, N. K. Nikol’skij
      Pages 207-252

  5. Back Matter

    Pages 261-268
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Keywords

About this book

The theory of generalized functions is a general method that makes it possible to consider and compute divergent integrals, sum divergent series, differentiate discontinuous functions, perform the operation of integration to any complex power and carry out other such operations that are impossible in classical analysis. Such operations are widely used in mathematical physics and the theory of differential equations, where the ideas of generalized func­ tions first arose, in other areas of analysis and beyond. The point of departure for this theory is to regard a function not as a mapping of point sets, but as a linear functional defined on smooth densi­ ties. This route leads to the loss of the concept of the value of function at a point, and also the possibility of multiplying functions, but it makes it pos­ sible to perform differentiation an unlimited number of times. The space of generalized functions of finite order is the minimal extension of the space of continuous functions in which coordinate differentiations are defined every­ where. In this sense the theory of generalized functions is a development of all of classical analysis, in particular harmonic analysis, and is to some extent the perfection of it. The more general theories of ultradistributions or gener­ alized functions of infinite order make it possible to consider infinite series of generalized derivatives of continuous functions.

Editors and Affiliations

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

    V. P. Havin

  • Department of Mathematics and Statistics, Mc Gill University, Montreal, Canada

    V. P. Havin

  • Steklov Mathematical Institute, St. Petersburg, Russia

    N. K. Nikol’skij

  • Département de Mathématiques, Université de Bordeaux I, Talence, Cedex, France

    N. K. Nikol’skij

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