Ebook: Fourier Analysis and Approximation: Vol. 1 One-Dimensional Theory
- Tags: Analysis
- Series: Mathematische Reihe 1
- Year: 1971
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-xvi
Preliminaries....Pages 1-24
Front Matter....Pages 25-28
Singular Integrals of Periodic Functions....Pages 29-93
Theorems of Jackson and Bernstein for Polynomials of Best Approximation and for Singular Integrals....Pages 94-118
Singular Integrals on the Line Group....Pages 119-161
Front Matter....Pages 163-166
Finite Fourier Transforms....Pages 167-187
Fourier Transforms Associated with the Line Group....Pages 188-230
Representation Theorems....Pages 231-277
Fourier Transform Methods and Second-Order Partial Differential Equations....Pages 278-302
Front Matter....Pages 303-304
Hilbert Transforms on the Real Line....Pages 305-333
Hilbert Transforms of Periodic Functions....Pages 334-354
Front Matter....Pages 355-356
Characterization in the Integral Case....Pages 357-390
Characterization in the Fractional Case....Pages 391-430
Front Matter....Pages 431-431
Saturation for Singular Integrals on X2? and Lp, 1 ? p ? 2....Pages 433-482
Saturation on X(?)....Pages 483-509
Back Matter....Pages 521-546
Content:
Front Matter....Pages i-xvi
Preliminaries....Pages 1-24
Front Matter....Pages 25-28
Singular Integrals of Periodic Functions....Pages 29-93
Theorems of Jackson and Bernstein for Polynomials of Best Approximation and for Singular Integrals....Pages 94-118
Singular Integrals on the Line Group....Pages 119-161
Front Matter....Pages 163-166
Finite Fourier Transforms....Pages 167-187
Fourier Transforms Associated with the Line Group....Pages 188-230
Representation Theorems....Pages 231-277
Fourier Transform Methods and Second-Order Partial Differential Equations....Pages 278-302
Front Matter....Pages 303-304
Hilbert Transforms on the Real Line....Pages 305-333
Hilbert Transforms of Periodic Functions....Pages 334-354
Front Matter....Pages 355-356
Characterization in the Integral Case....Pages 357-390
Characterization in the Fractional Case....Pages 391-430
Front Matter....Pages 431-431
Saturation for Singular Integrals on X2? and Lp, 1 ? p ? 2....Pages 433-482
Saturation on X(?)....Pages 483-509
Back Matter....Pages 521-546
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