Fourier Analysis and Approximation

1. Front Matter

   Pages i-xvi

   PDF

2. Preliminaries

   1. Preliminaries

      • Paul L. Butzer, Rolf J. Nessel

      Pages 1-24

3. Approximation by Singular Integrals

   1. Front Matter

      Pages 25-28

      PDF

   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

      PDF

   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

      PDF

   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

      PDF

   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

      PDF

   2. Saturation for Singular Integrals on X_2π and L^p, 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

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.

• Fourier series
• Fourier transform
• Singular integral
• calculus
• differential equation
• fourier analysis
• harmonic analysis
• integral transform

• Rheinisch Westfälische Technische Hochschule, Aachen, Germany

  Paul L. Butzer, Rolf J. Nessel

Policies and ethics