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Birkhäuser

Fourier Analysis and Approximation

One Dimensional Theory

  • Book
  • © 1971

Overview

Authors:
  1. Paul L. Butzer

    1. Rheinisch Westfälische Technische Hochschule, Aachen, Germany

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  2. Rolf J. Nessel

    1. Rheinisch Westfälische Technische Hochschule, Aachen, Germany

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften (LMW, volume 1)

Part of the book sub series: Mathematische Reihe (LMW/MA)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Preliminaries

    1. Preliminaries

      • Paul L. Butzer, Rolf J. Nessel
      Pages 1-24

  3. Approximation by Singular Integrals

    1. Front Matter

      Pages 25-28
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    2. Singular Integrals of Periodic Functions

      • Paul L. Butzer, Rolf J. Nessel
      Pages 29-93

    3. Theorems of Jackson and Bernstein for Polynomials of Best Approximation and for Singular Integrals

      • Paul L. Butzer, Rolf J. Nessel
      Pages 94-118

    4. Singular Integrals on the Line Group

      • Paul L. Butzer, Rolf J. Nessel
      Pages 119-161

  4. Fourier Transforms

    1. Front Matter

      Pages 163-166
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    2. Finite Fourier Transforms

      • Paul L. Butzer, Rolf J. Nessel
      Pages 167-187

    3. Fourier Transforms Associated with the Line Group

      • Paul L. Butzer, Rolf J. Nessel
      Pages 188-230

    4. Representation Theorems

      • Paul L. Butzer, Rolf J. Nessel
      Pages 231-277

    5. Fourier Transform Methods and Second-Order Partial Differential Equations

      • Paul L. Butzer, Rolf J. Nessel
      Pages 278-302

  5. Hilbert Transforms

    1. Front Matter

      Pages 303-304
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    2. Hilbert Transforms on the Real Line

      • Paul L. Butzer, Rolf J. Nessel
      Pages 305-333

    3. Hilbert Transforms of Periodic Functions

      • Paul L. Butzer, Rolf J. Nessel
      Pages 334-354

  6. Characterization of Certain Function Classes

    1. Front Matter

      Pages 355-356
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    2. Characterization in the Integral Case

      • Paul L. Butzer, Rolf J. Nessel
      Pages 357-390

    3. Characterization in the Fractional Case

      • Paul L. Butzer, Rolf J. Nessel
      Pages 391-430

  7. Saturation Theory

    1. Front Matter

      Pages 431-431
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    2. Saturation for Singular Integrals on X2π and Lp, 1 ≤ p ≤ 2

      • Paul L. Butzer, Rolf J. Nessel
      Pages 433-482

    3. Saturation on X(ℝ)

      • Paul L. Butzer, Rolf J. Nessel
      Pages 483-509
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Keywords

About this book

At the international conference on 'Harmonic Analysis and Integral Transforms', conducted by one of the authors at the Mathematical Research Institute in Oberwolfach (Black Forest) in August 1965, it was felt that there was a real need for a book on Fourier analysis stressing (i) parallel treatment of Fourier series and Fourier trans­ forms from a transform point of view, (ii) treatment of Fourier transforms in LP(lRn)_ space not only for p = 1 and p = 2, (iii) classical solution of partial differential equations with completely rigorous proofs, (iv) theory of singular integrals of convolu­ tion type, (v) applications to approximation theory including saturation theory, (vi) multiplier theory, (vii) Hilbert transforms, Riesz fractional integrals, Bessel potentials, (viii) Fourier transform methods on locally compact groups. This study aims to consider these aspects, presenting a systematic treatment of Fourier analysis on the circle as well as on the infinite line, and of those areas of approximation theory which are in some way or other related thereto. A second volume is in preparation which goes beyond the one-dimensional theory presented here to cover the subject for functions of several variables. Approximately a half of this first volume deals with the theories of Fourier series and of Fourier integrals from a transform point of view.

Authors and Affiliations

  • Rheinisch Westfälische Technische Hochschule, Aachen, Germany

    Paul L. Butzer, Rolf J. Nessel

Bibliographic Information

  • Book Title: Fourier Analysis and Approximation

  • Book Subtitle: One Dimensional Theory

  • Authors: Paul L. Butzer, Rolf J. Nessel

  • Series Title: Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-0348-7448-9

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Birkhäuser Verlag Basel 1971

  • Softcover ISBN: 978-3-0348-7450-2Published: 03 May 2012

  • eBook ISBN: 978-3-0348-7448-9Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: XVI, 553

  • Topics: Analysis

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