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A treatise on trigonometric series. Volume 1 /
A Treatise on Trigonometric Series.
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés Ruso |
| Publicado: |
New York :
The Macmillan Company,
1964.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Front Cover; A Treatise on Trigonometric Series; Copyright Page; Table of Contents; CONTENTS OF VOLUME II; TRANSLATOR'S PREFACE; AUTHOR'S PREFACE; NOTATION; INTRODUCTORY MATERIAL; 1. ANALYTICAL THEOREMS; II. NUMERICAL SERIES, SUMMATION; III. INEQUALITIES FOR NUMBERS, SERIES AND INTEGRALS; IV. THEORY OF SETS AND THEORY OF FUNCTIONS; V. FUNCTIONAL ANALYSIS; VI. THEORY OF APPROXIMATION OF FUNCTIONS BYTRIGONOMETRIC POLYNOMIALS; CHAPTER I. BASIC CONCEPTS AND THEOREMS IN THETHEORY OF TRIGONOMETRIC SERIES; 1. The concept of a trigonometric series; conjugate series.
- 2. The complex form of a trigonometric series 3. A brief historical synopsis; 4. Fourier formulae; 5. The complex form of a Fourier series; 6. Problems in the theory of Fourier series; Fourier-Lebesgue series; 7. Expansion into atrigonometric series of a function with period 2l; 8. Fourier series for even and odd functions; 9. Fourier series with respect to the orthogonal system; 10. Completeness of an orthogonal system; 11. Completeness of the trigonometric system in the space L; 12. Uniformly convergent Fourier series.
- 13. The minimum property of the partial sums of a Fourier series Bessel's inequality; 14. Convergence of a Fourier series in the metric space L2; 15. Concept of the closure of the system. Relationship between closure and completeness; 16. The Riesz-Fischer theorem; 17. The Riesz-Fischer theorem and Parseval's equality for a trigonometric system; 18. Parseval's equality for the product of two functions; 19. The tending to zero of Fourier coefficients; 20. Fej�er's lemma; 21. Estimate of Fourier coefficients in terms of the integral modulus of continuity of the function.
- 22. Fourier coefficients for functions of bounded variation 23. Formal operations on Fourier series; 24. Fourier series for repeatedly differentiated functions; 25. On Fourier coefficients for analytic functions; 26. The simplest cases of absolute and uniform convergence of Fourier series; 27. Weierstrass's theorem on the approximation of a continuous function by trigonometric polynomials; 28. The density of a class of trigonometric polynomials in the spaces LP (p> 1); 29. Dirichlet's kernel and its conjugate kernel.
- 30. Sine or cosine series with monotonically decreasing coefficients 31. Integral expressions for the partial sums of a Fourier series and its conjugate series; 32. Simplification of expressions for Sn(x) and Sn(x); 33. Riemann's principle of localization; 34. Steinhauses theorem; 35. Integral dx. Lebesgue constants; 36. Estimate of the partial sums of a Fourier series of a bounded function; 37. Criterion of convergence of a Fourier series; 38. Dini's test; 39. Jordan's test; 40. Integration of Fourier series; 41. Gibbs's phenomenon; 42. Determination of the magnitude of the discontinuity of a function from its Fourier series.