Kawata / Birnbaum / Lukacs Fourier Analysis in Probability Theory

1. Auflage 2014
ISBN: 978-1-4832-1852-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 680 Seiten, Web PDF

Fourier Analysis in Probability Theory provides useful results from the theories of Fourier series, Fourier transforms, Laplace transforms, and other related studies. This 14-chapter work highlights the clarification of the interactions and analogies among these theories. Chapters 1 to 8 present the elements of classical Fourier analysis, in the context of their applications to probability theory. Chapters 9 to 14 are devoted to basic results from the theory of characteristic functions of probability distributors, the convergence of distribution functions in terms of characteristic functions, and series of independent random variables. This book will be of value to mathematicians, engineers, teachers, and students.

Autoren/Hrsg.

Kawata, Tatsuo

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Fourier Analysis in Probability Theory;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;12
6;Chapter I. Introduction;14
6.1;1.1. Measurable Space; Probability Space;14
6.2;1.2. Measurable Functions; Random Variables;19
6.3;1.3. Product Space;20
6.4;1.4. Integrals;21
6.5;1.5. The Fubini–Tonelli Theorem;25
6.6;1.6. Integrals on R;26
6.7;1.7. Functions of Bounded Variation;29
6.8;1.8. Signed Measure; Decomposition Theorems;31
6.9;1.9. The Lebesgue Integral on R;33
6.10;1.10. Inequalities;37
6.11;1.11. Convex Functions;40
6.12;1.12. Analytic Functions;41
6.13;1.13. Jensen's and Carleman's Theorems;44
6.14;1.14. Analytic Continuation;45
6.15;1.15. Maximum Modulus Theorem and Theorems of Phragmén–Lindelöf;48
6.16;1.16. Inner Product Space;50
7;Chapter Il. Fourier Series and Fourier Transforms;56
7.1;2.1. The Riemann–Lebesgue Lemma;56
7.2;2.2. Fourier Series;62
7.3;2.3. The Fourier Transform of a Function in L1(—8,8);65
7.4;2.4. Magnitude of Fourier Coefficients; the Continuity Modulus;66
7.5;2.5. More about the Magnitude of Fourier Coefficients;69
7.6;2.6. Some Elementary Lemmas;72
7.7;2.7. Continuity and Magnitude of Fourier Transforms;75
7.8;2.8. Operations on Fourier Series;77
7.9;2.9. Operations on Fourier Transforms;80
7.10;2.10. Completeness of Trigonometric Functions;82
7.11;2.11. Unicity Theorem for Fourier Transforms;86
7.12;2.12. Fourier Series and Fourier Transform of Convolutions;87
7.13;NOTES;91
8;Chapter Ill. Fourier–Stieltjes Coefficients, Fourier–Stieltjes Transforms and Characteristic Functions;94
8.1;3.1. Monotone Functions and Distribution Functions;94
8.2;3.2. Fourier–Stieltjes Series;97
8.3;3.3. Average of Fourier–Stieltjes Coefficients;99
8.4;3.4. Unicity Theorem for Fourier–Stieltjes Coefficients;101
8.5;3.5. Fourier–Stieltjes Transform and Characteristic Function;102
8.6;3.6. Periodic Characteristic Functions;106
8.7;3.7. Some Inequality Relations for Characteristic Functions;108
8.8;3.8. Average of a Characteristic Function;115
8.9;3.9. Convolution of Nondecreasing Functions;117
8.10;3.10. The Fourier–Stieltjes Transform of a Convolution and the Bernoulli Convolution;121
8.11;NOTES;124
9;Chapter IV. Convergence and Summability Theorems;126
9.1;4.1. Convergence of Fourier Series;126
9.2;4.2. Convergence of Fourier–Stieltjes Series;134
9.3;4.3. Fourier's Integral Theorems; Inversion Formulas for Fourier Transforms;136
9.4;4.4. Inversion Formula for Fourier–Stieltjes Transforms;141
9.5;4.5. Summability;144
9.6;4.6. (C, 1)-Summability for Fourier Series;148
9.7;4.7. Abel-Summability for Fourier Series;153
9.8;4.8. Summability Theorems for Fourier Transforms;158
9.9;4.9. Determination of the Absolutely Continuous Component of a Nondecreasing Function;164
9.10;4.10. Fourier Series and Approximate Fourier Series of a Fourier–Stieltjes Transform;167
9.11;4.11. Some Examples, Using Fourier Transforms;171
9.12;NOTES;177
10;Chapter V. General Convergence Theorems;179
10.1;5.1. Nature of the Problems;179
10.2;5.2. Some General Convergence Theorems I;180
10.3;5.3. Some General Convergence Theorems II;187
10.4;5.4. General Convergence Theorems for the Stieltjes Integral;191
10.5;5.5. Wiener's Formula;194
10.6;5.6. Applications of General Convergence Theorems to the Estimates of a Distribution Function;199
10.7;NOTES;205
11;Chapter VI. L2-Theory of Fourier Series and Fourier Transforms;207
11.1;6.1. Fourier Series in an Inner Product Space;207
11.2;6.2. Fourier Transform of a Function in L2(—8,8);214
11.3;6.3. The Class H2 of Analytic Functions;223
11.4;6.4. A Theorem of Szegö and Smirnov;227
11.5;6.5. The Class £2 of Analytic Functions;231
11.6;6.6. A Theorem of Paley and Wiener;241
11.7;NOTES;244
12;Chapter VII. Laplace and Mellin Transforms;245
12.1;7.1. The Laplace Transform;245
12.2;7.2. The Convergence Abscissa;251
12.3;7.3. Analyticity of a Laplace–Stieltjes Transform;255
12.4;7.4. Inversion Formulas for Laplace Transforms;259
12.5;7.5. The Laplace Transform of a Convolution;265
12.6;7.6. Operations on Laplace Transforms and Some Examples;272
12.7;7.7. The Bilateral Laplace–Stieltjes Transform;278
12.8;7.8. Mellin–Stieltjes Transforms;282
12.9;7.9. The Mellin Transform;286
12.10;NOTES;289
13;Chapter VIII. More Theorems on Fourier and Laplace Transforms;291
13.1;8.1. A Theorem of Hardy;291
13.2;8.2. A Theorem of Paley and Wiener on Exponential Entire Functions;297
13.3;8.3. Theorems of Ingham and Levinson;301
13.4;8.4. Singularities of Laplace Transforms;311
13.5;8.5. Abelian Theorems for Laplace Transforms;314
13.6;8.6. Tauberian Theorems;317
13.7;8.7. Multiple Fourier Series and Transforms;327
13.8;8.8. Nondecreasing Functions and Distribution Functions in Rm;336
13.9;8.9. The Multiple Fourier–Stieltjes Transform;339
13.10;NOTES;341
14;Chapter IX. Convergence of Distribution Functions and Characteristic Functions;343
14.1;9.1. Helly Theorems and Convergence of Nondecreasing Functions;343
14.2;9.2. Convergence of Distribution Functions with Bounded Spectra;352
14.3;9.3. Convergence of Distribution Functions;354
14.4;9.4. Continuous Distribution Functions: A General Integral Transform of a Characteristic Function;366
14.5;9.5. A Basic Theorem on Analytic Characteristic Functions;369
14.6;9.6. Continuity Theorems on Intervals and Uniqueness Theorems;370
14.7;9.7. The Compact Set of Characteristic Functions;375
14.8;NOTES;378
15;Chapter X. Some Properties of Characteristic Functions;379
15.1;10.1. Characteristic Properties of Fourier Coefficients;379
15.2;10.2. Basic Theorems on Characterization of a Characteristic Function;385
15.3;10.3. Characteristic Properties of Characteristic Functions;389
15.4;10.4. Functions of the Wiener Class;396
15.5;10.5. Some Sufficient Criteria for Characteristic Functions;398
15.6;10.6. More Criteria for Characteristic Functions;405
15.7;NOTES;410
16;Chapter XI. Distribution Functions and Their Characteristic Functions;413
16.1;11.1. Moments, Basic Properties;413
16.2;11.2. Smoothness of a Characteristic Function and the Existence of Moments;421
16.3;11.3. More about Smoothness of Characteristic Functions and Existence of Moments;432
16.4;11.4. Absolute Moments;441
16.5;11.5. Boundedness of the Spectra of Distribution Functions;444
16.6;11.6. Integrable Characteristic Functions;450
16.7;11.7. Analyticity of Distribution Functions;452
16.8;11.8. Mean Concentration Function of a Distribution Function;458
16.9;11.9. Some Properties of Analytic Characteristic Functions;465
16.10;11.10. Characteristic Functions Analytic in the Half-Plane;469
16.11;11.11. Entire Characteristic Functions I;473
16.12;11.12. Entire Characteristic Functions II;479
16.13;NOTES;486
17;Chapter XII. Convergence of Series of Independent Random Variables;488
17.1;12.1. Convergence of a Sequence of Random Variables;488
17.2;12.2. The Borel Theorem;499
17.3;12.3. The Zero–One Law;501
17.4;12.4. The Equivalence Theorem;505
17.5;12.5. The Three Series Theorem;509
17.6;12.6. Sufficient Conditions for the Convergence of a Series;514
17.7;12.7. Convergence Criteria and the Typical Function;517
17.8;12.8. Rademacher and Steinhaus Functions;520
17.9;12.9. Convergence of Products of Characteristic Functions;526
17.10;12.10. Unconditional Convergence;532
17.11;12.11. Absolute Convergence;539
17.12;12.12. Essential Convergence;543
17.13;NOTES;547
18;Chapter XIII. Properties of Sums of Independent Random Variables; Convergence of Series in the Mean;549
18.1;13.1. Continuity and Discontinuity Properties of the Sum of a Series;549
18.2;13.2. Integrability of the Sum of a Series;556
18.3;13.3. Magnitude of the Characteristic Functions of the Sums of Series;559
18.4;13.4. Distribution Functions of the Sums of Rademacher Series; Characteristic Functions of Singular Distributions;569
18.5;13.5. Further Theorems on Rademacher Series;578
18.6;13.6. Sums of Independent Random Variables;586
18.7;13.7. Convergent Systems;596
18.8;13.8. Integrability of Sums of Series; Strong and Weak Convergences of Series;599
18.9;13.9. Vanishing of the Sum of a Series;603
18.10;13.10. Summability of Series;609
18.11;NOTES;615
19;Chapter XIV. Some Special Series of Random Variables;618
19.1;14.1. Fourier Series with Rademacher Coefficients;618
19.2;14.2. Random Fourier Series;625
19.3;14.3. Random Power Series, Convergence;632
19.4;14.4. Convergence of Random Power Series with Identically and Independently Distributed Random Coefficients;638
19.5;14.5. Analytic Continuation of Random Power Series;641
19.6;14.6. Fourier Series with Orthogonal Random Coefficients;647
19.7;NOTES;653
20;References;655
21;Index;674