Fourier Analysis in Probability Theory

PrefaceI Introduction 1.1 Measurable Space; Probability Space 1.2 Measurable Functions; Random Variables 1.3 Product Space 1.4 Integrals 1.5 The Fubini-Tonelli Theorem 1.6 Integrals on R1 1.7 Functions of Bounded Variation 1.8 Signed Measure; Decomposition Theorems 1.9 The Lebesgue Integral on R1 1.10 Inequalities 1.11 Convex Functions 1.12 Analytic Functions 1.13 Jensen's and Carleman's Theorems 1.14 Analytic Continuation 1.15 Maximum Modulus Theorem and Theorems of Phragmén-Lindelöf 1.16 Inner Product SpaceII Fourier Series and Fourier Transforms 2.1 The Riemann-Lebesgue Lemma 2.2 Fourier Series 2.3 The Fourier Transform of a Function in L1(-8, 8) 2.4 Magnitude of Fourier Coefficients; the Continuity Modulus 2.5 More About the Magnitude of Fourier Coefficients 2.6 Some Elementary Lemmas 2.7 Continuity and Magnitude of Fourier Transforms 2.8 Operations on Fourier Series 2.9 Operations on Fourier Transforms 2.10 Completeness of Trigonometric Functions 2.11 Unicity Theorem for Fourier Transforms 2.12 Fourier Series and Fourier Transform of Convolutions NotesIII Fourier-Stieltjes Coefficients, Fourier-Stieltjes Transforms and Characteristic Functions 3.1 Monotone Functions and Distribution Functions 3.2 Fourier-Stieltjes Series 3.3 Average of Fourier-Stieltjes Coefficients 3.4 Unicity Theorem for Fourier-Stieltjes Coefficients 3.5 Fourier-Stieltjes Transform and Characteristic Function 3.6 Periodic Characteristic Functions 3.7 Some Inequality Relations for Characteristic Functions 3.8 Average of a Characteristic Function 3.9 Convolution of Nondecreasing Functions 3.10 The Fourier-Stieltjes Transform of a Convolution and the Bernoulli Convolution NotesIV Convergence and Summability Theorems 4.1 Convergence of Fourier Series 4.2 Convergence of Fourier-Stieltjes Series 4.3 Fourier's Integral Theorems; Inversion Formulas for Fourier Transforms 4.4 Inversion Formula for Fourier-Stieltjes Transforms 4.5 Summability 4.6 (C,1)-Summability for Fourier Series 4.7 Abel-Summability for Fourier Series 4.8 Summability Theorems for Fourier Transforms 4.9 Determination of the Absolutely Continuous Component of a Nondecreasing Function 4.10 Fourier Series and Approximate Fourier Series of a Fourier-Stieltjes Transform 4.11 Some Examples, Using Fourier Transforms NotesV General Convergence Theorems 5.1 Nature of the Problems 5.2 Some General Convergence Theorems I 5.3 Some General Convergence Theorems II 5.4 General Convergence Theorems for the Stieltjes Integral 5.5 Wiener's Formula 5.6 Applications of General Convergence Theorems to the Estimates of a Distribution Function NotesVI L2-Theory of Fourier Series and Fourier Transforms 6.1 Fourier Series in an Inner Product Space 6.2 Fourier Transform of a Function in L2(-8, 8) 6.3 The Class H2 of Analytic Functions 6.4 A Theorem of Szegö and Smirnov 6.5 The Class ¿2 of Analytic Functions 6.6 A Theorem of Paley and Wiener NotesVII Laplace and Mellin Transforms 7.1 The Laplace Transform 7.2 The Convergence Abscissa 7.3 Analyticity of a Laplace-Stieltjes Transform 7.4 Inversion Formulas for Laplace Transforms 7.5 The Laplace Transform of a Convolution 7.6 Operations of Laplace Transforms and Some Examples 7.7 The Bilateral Laplace-Stieltjes Transform 7.8 Mellin-Stieltjes Transforms 7.9 The Mellin Transform NotesVIII More Theorems on Fourier and Laplace Transforms 8.1 A Theorem of Hardy 8.2 A Theorem of Paley and Wiener on Exponential Entire Functions 8.3 Theorems of Ingham and Levinson 8.4 Singularities of Laplace Transforms 8.5 Abelian Theorems for Laplace Transforms 8.6 Tauberian Theorems 8.7 Multiple Fourier Series and Transforms 8.