# A Treatise on Trigonometric Series

## 1st Edition

### Volume 1

**Authors:**N. K. Bary

**eBook ISBN:**9781483224190

**Imprint:**Pergamon

**Published Date:**1st January 1964

**Page Count:**578

## Description

A Treatise on Trigonometric Series, Volume 1 deals comprehensively with the classical theory of Fourier series. This book presents the investigation of best approximations of functions by trigonometric polynomials.

Organized into six chapters, this volume begins with an overview of the fundamental concepts and theorems in the theory of trigonometric series, which play a significant role in mathematics and in many of its applications. This text then explores the properties of the Fourier coefficient function and estimates the rate at which its Fourier coefficients tend to zero. Other chapters consider some tests for the convergence of a Fourier series at a given point. This book discusses as well the conditions under which the series does converge uniformly. The final chapter deals with adjustment of a summable function outside a given perfect set.

This book is a valuable resource for advanced students and research workers. Mathematicians will also find this book useful.

## Table of Contents

Contents of Volume II

Translator's Preface

Author's Preface

Notation

Introductory Material

I. Analytical Theorems

1. Abel's Transformation

2. Second Mean Value Theorem

3. Convex Curves and Convex Sequences

II. Numerical Series, Summation

4. Series with Monotonically Decreasing Terms

5. Linear Methods of Summation

6. Method of Arithmetic Means [or (C, 1)]

7. Abel's Method

III. Inequalities for Numbers, Series and Integrals

8. Numerical Inequalities

9. Holder's Inequality

10. Minkowski's Inequality

11. O- and o-Relationships for Series and Integrals

IV. Theory of Sets and Theory of Functions

12. on the Upper Limit of a Sequence of Sets

13. Convergence in Measure

14. Passage to The Limit Under Lebesgue's Integral Sign

15. Lebesgue Points

16. Riemann-Stieltjes Integral

17. Helly's Two Theorems

18. Fubini's Theorem

V. Functional Analysis

19. Linear Functionals in C

20. Linear Functionals in Lp(p > 1)

21. Convergence in Norm in the Spaces Lp

VI. Theory of Approximation of Functions by Trigonometric Polynomials

22. Elementary Properties of Trigonometric Polynomials

23. Bernstein's Inequality

24. Trigonometric Polynomial of Best Approximation

25. Modulus of Continuity, Modulus of Smoothness, and Integral Modulus of Continuity

Chapter I. Basic Concepts and Theorems in the Theory 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 a Trigonometric 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. Fejér'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. Steinhaus's Theorem

35. Integral ∞∫0[(sinx)/x] 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

43. Singularities of Fourier Series of Continuous Functions. Fejér Polynomials

44. A Continuous Function with a Fourier Series Which Converges Everywhere But Not Uniformly

45. Continuous Function with a Fourier Series Divergent at One Point (Fejér's Example)

46. Divergence at One Point (Lebesgue's Example)

47. Summation of a Fourier Series by Fejér's Method

48. Corollaries of Fejér's Theorem

49. Fejér-Lebesgue Theorem

50. Estimate of the Partial Sums of a Fourier Series

51. Convergence Factors

52. Comparison of Dirichlet and Fejér Kernels

53. Summation of Fourier Series by the Abel-Poisson Method

54. Poisson Kernel and Poisson Integral

55. Behaviour of the Poisson Integral at Points of Continuity of a Function

56. Behaviour of a Poisson Integral in the General Case

57. The Dirichlet Problem

58. Summation by Poisson's Method of a Differentiated Fourier Series

59. Poisson-Stieltjes Integral

60. Fejér and Poisson Sums for Different Classes of Functions

61. General Trigonometric Series. The Lusin-Denjoy Theorem

62. The Cantor-Lebesgue Theorem

63. an Example of an Everywhere Divergent Series with Coefficients Tending to Zero

64. A Study of the Convergence of One Class of Trigonometric Series

65. Lacunary Sequences and Lacunary Series

66. Smooth Functions

67. The Schwarz Second Derivative

68. Riemann's Method of Summation

69. Application of Riemann's Method of Summation to Fourier Series

70. Cantor's Theorem of Uniqueness

71. Riemann's Principle of Localization for General Trigonometric Series

72. Du Bois-Reymond's Theorem

73. Problems

Chapter II. Fourier Coefficients

1. Introduction

2. The Order of Fourier Coefficients for Functions of Bounded Variation. Criterion for the Continuity of Functions of Bounded Variation

3. Concerning Fourier Coefficients for Functions of the Class Lip α

4. The Relationship Between the Order of Summability of a Function and the Fourier Coefficients

5. The Generalization of Parseval's Equality for the Product of Two Functions

6. The Rate at Which the Fourier Coefficients of Summable Functions Tend to Zero

7. Auxiliary Theorems Concerning The Rademacher System

8. Absence of Criteria Applicable to the Moduli of Coefficients

9. Some Necessity Conditions for Fourier Coefficients

10. Salem's Necessary and Sufficient Conditions

11. The Trigonometric Problem of Moments

12. Coefficients of Trigonometric Series with Non-Negative Partial Sums

13. Transformation of Fourier Series

14. Problems

Chapter III. The Convergence of a Fourier Series at a Point

1. Introduction

2. Comparison of the Dini and Jordan Tests

3. The De La Vallée-Poussin Test and its Comparison with the Dini and Jordan Tests

4. The Young Test

5. The Relationship Between the Young Test and the Dini, Jordan and De La Vallée-Poussin Tests

6. The Lebesgue Test

7. A Comparison of the Lebesgue Test with All the Preceding Tests

8. The Lebesgue-Gergen Test

9. Concerning The Necessity Conditions for Convergence at a Point

10. Sufficiency Convergence Tests at a Point with Additional Restrictions on the Coefficients of the Series

11. A Note Concerning the Uniform Convergence of a Fourier Series in Some Interval

12. Problems

Chapter IV. Fourier Series of Continuous Functions

1. Introduction

2. Sufficiency Conditions for Uniform Convergence, Expressed in Terms of Fourier Coefficients

3. Sufficiency Conditions for Uniform Convergence in Terms of the Best Approximations

4. The Dini-Lipschitz Test 301

5. The Salem Test. Functions of φ-Bounded Variation

6. The Rogosinski Identity

7. A Test of Uniform Convergence, Using the Integrated Series

8. The Generalization of the Dini-Lipschitz Test (in the Integral Form)

9. Uniform Convergence Over the Interval [a, b]

10. The Sâto Test

11. Concerning Uniform Convergence Near Every Point of an Interval

12. Concerning Operations on Functions to Obtain Uniformly Convergent Fourier Series

13. Concerning Uniform Convergence by Rearrangement of the Signs in the Terms of the Series

14. Extremal Properties of Some Trigonometric Polynomials

15. The Choice of Arguments for Given Moduli of The Terms of the Series

16. Concerning Fourier Coefficients of Continuous Functions

17. Concerning the Singularities of Fourier Series of Continuous Functions

18. A Continuous Function with a Fourier Series Non-Uniformly Convergent in Any Interval

19. Concerning a Set of Points of Divergence for a Trigonometric Series

20. A Continuous Function with a Fourier Series Divergent in a Set of The Power of the Continuum

21. Divergence in a Given Denumerable Set

22. Divergence in a Set of The Power of The Continuum for Bounded Partial Sums

23. Divergence for a Series of f2(x)

24. Sub-Sequences of Partial Sums of Fourier Series for Continuous Functions

25. Resolution Into the Sum of Two Series Convergent in Sets of Positive Measure

26. Problems

Chapter V. Convergence and Divergence of a Fourier Series in a Set

1. Introduction

2. The Kolmogorov-Seliverstov and Plessner Theorem

3. A Convergence Test Expressed by the First Differences of the Coefficients

4. Convergence Factors 370

5. Other Forms of the Condition Imposed in the Kolmogorov-Seliverstov and Plessner Theorem

6. Corollaries of Plessner's Theorem

7. Concerning the Equivalence of Some Conditions Expressed in Terms of Integrals and in Terms of Series

8. A Test of Almost Everywhere Convergence for Functions of Lp(1 ≤ P ≤ 2)

9. Expression of the Conditions of Almost Everywhere Convergence in Terms of the Quadratic Moduli of Continuity and the Best Approximations

10. Tests of Almost Everywhere Convergence in an Interval of Length Less than 2π

11. Indices of Convergence

12. The Convex Capacity of Sets

13. A Convergence Test, Using an Integrated Series

14. The Salem Test

15. The Marcinkiewicz Test

16. Convergence Test Expressed by the Logarithmic Measure of the Set

17. Fourier Series, Almost Everywhere Divergent

18. The Impossibility of Strengthening the Marcinkiewicz Test

19. Concerning the Series Conjugate to an Almost Everywhere Divergent Fourier Series

20. A Fourier Series, Divergent at Every Point

21. Concerning the Principle of Localization for Sets

22. Concerning the Convergence of a Fourier Series in a Given Set and its Divergence Outside it

23. The Problem of Convergence and the Principle of Localization for Fourier Series with Rearranged Terms

24. Problems

Chapter VI. "Adjustment" of Functions in a Set of Small Measure

1. Introduction

2. Two Elementary Lemmas

3. Lemma Concerning the Dirichlet Factor

4. "Adjustment" of a Function to Obtain a Uniformly Convergent Fourier Series

5. The Strengthened C-Property

6. Problems Connected with the "Adjustment" of Functions

7. "Adjustment" of a Summable Function Outside a Given Perfect Set

8. Problems

Appendix to Chapter II

1. The Phragmén-Lindelof Principle

2. Modulus of Continuity and Modulus of Smoothness in Lp(P ≥ 1)

3. A Converse of the Holder Inequality

4. The Banach-Steinhaus Theorem

To Chapter IV

5. Categories of Sets

6. Riemann's and Carathéodory's Theorems

7. The Connection Between the Modulus of Continuity and the Best Approximation of a Function

To Chapter V

8. μ-Measures and Integrals

Bibliography

Index

## Details

- No. of pages:
- 578

- Language:
- English

- Copyright:
- © Pergamon 1964

- Published:
- 1st January 1964

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483224190