A Treatise on Trigonometric Series

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.

Contents of Volume IITranslator's PrefaceAuthor's PrefaceNotationIntroductory 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 ContinuityChapter 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. ProblemsChapter 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. ProblemsChapter 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. ProblemsChapter 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. ProblemsChapter 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. ProblemsChapter 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. ProblemsAppendix 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 TheoremTo 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 FunctionTo Chapter V 8. μ-Measures and IntegralsBibliographyIndex

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