The Hardy-Littlewood Theorems 1 Introduction 2 Examples of Summability Methods Abelian Theorems and Tauberian Question 3 Simple Applications of Cesa(')ro, Abel and Borel Summability 4 Lambert Summability in Number Theory 5 Tauber's Theorems for Abel Summability 6 Tauberian Theorem for Cesa(')ro Summability 7 Hardy-Littlewood Tauberians for Abel Summability 8 Tauberians Involving Dirichlet Series 9 Tauberians for Borel Summability 10 Lambert Tauberian and Prime Number Theorem 11 Karamata's Method for Power Series 12 Wielandt's Variation on the Method 13 Transition from Series to Integrals 14 Extension of Tauber's Theorems to Laplace-Stieltjes Transforms 15 Hardy-Littlewood Type Theorems Involving Laplace Transforms 16 Other Tauberian Conditions: Slowly Decreasing Functions 17 Asymptotics for Derivatives 18 Integral Tauberians for Cesa(')ro Summability 19 The Method of the Monotone Minorant 20 Boundedness Theorem Involving a General-Kernel Transform 21 Laplace-Stieltjes and Stieltjes Transform 22 General Dirichlet Series 23 The High-Indices Theorem 24 Optimality of Tauberian Conditions 25 Tauberian Theorems of Nonstandard Type 26 Important Properties of the Zeta Function Wiener's Theory 1 Introduction 2 Wiener Problem: Pitt's Form 3 Testing Equation for Wiener Kernels 4 Original Wiener Problem 5 Wiener's Theorem With Additions by Pitt 6 Direct Applications of the Testing Equations 7 Fourier Analysis of Wiener Kernels 8 The Principal Wiener Theorems 9 Proof of the Division Theorem 10 Wiener Families of Kernels 11 Distributional Approach to Wiener Theory 12 General Tauberian for Lambert SummabilitY 13 Wiener's 'Second Tauberian Theorem' 14 A Wiener Theorem for Series 15 Extensions 16 Discussion of the Tauberian Conditions 17 Landau-Ingham Asymptotics 18 Ingham Summability 19 Application of Wiener Theory to Harmonic Functions Ⅲ Complex Tauberian Theorems 1 Introduction 2 A Landau-Type Tauberian for Dirichlet Series 3 Mellin Transforms 4 The Wiener-Ikehara Theorem 5 Newer Approach to Wiener-Ikehara 6 Newman's Way to the PNT. Work of Ingham 7 Laplace Transforms of Bounded Functions 8 Application to Dirichlet Series and the PNT 9 Laplace Transforms of Functions Bounded From Below 10 Tauberian Conditions Other Than Boundedness 11 An Optimal Constant in Theorem 10.1 12 Fatou and Riesz. General Dirichlet Series 13 Newer Extensions of Fatou-Riesz 14 Pseudofunction Boundary Behavior 15 Applications to Operator Theory 16 Complex Remainder Theory 17 The Remainder in Fatou's Theorem 18 Remainders in Hardy-Littlewood Theorems Involving Power Series 19 A Remainder for the Stieltjes Transform Ⅳ Karamata's Heritage: Regular Variation 1 Introduction 2 Slow and Regular Variation 3 Proof of the Basic Properties 4 Possible Pathology 5 Karamata's Characterization of Regularly. Varying Functions 6 Related Classes of Functions 7 Integral Transforms and Regular Variation: Introduction 8 Karamata's Theorem for Laplace Transforms 9 Stieltjes and Other Transforms 10 The Ratio Theorem 11 Beurling Slow Variation 12 A Result in Higher-Order Theory 13 Mercerian Theorems 14 Proof of Theorem 13.2 15 Asymptotics Involving Large Laplace Transforms 16 Transforms of Exponential Growth: Logarithmic Theory 17 Strong Asymptotics: General Case 18 Application to Exponential Growth 19 Very Large Laplace Transforms 20 Logarithmic Theory for Very Large Transforms 21 Large Transforms: Complex Approach 22 Proof of Proposition 21.4 23 Asymptotics for Partitions 24 Two-Sided Laplace Transforms Extensions of the Classical Theory 1 Introduction 2 Preliminaries on Banach Algebras 3 Algebraic Form of Wiener's Theorem 4 Weighted L1 Spaces 5 Gelfand's Theory of Maximal Ideals 6 Application to the Banach Algebra Aω = (Lω, C) 7 Regularity Condition for Lω 8 The Closed Maximal Ideals in Lω 9 Related Questions Involving Weighted Spaces 10 A Boundedness Theorem of Pitt 11 Proof of Theorem 10.2, Part 1 12 Theorem 10.2: Proof that S(y) = Q(eεY) 13 Theorem 10.2: Proof that S(y) = Q{eφ(y) 14 Boundedness Through Functional Analysis 15 Limitable Sequences as Elements of an FK-space 16 Perfect Matrix Methods 17 Methods with Sectional Convergence 18 Existence of (Limitable) Bounded Divergent Sequences 19 Bounded Divergent Sequences, Continued 20 Gap Tauberian Theorems 21 The Abel Method 22 Recurrent Events 23 The Theorem of Erd6s, Feller and Pollard 24 Milin's Theorem 25 Some Propositions 26 Proof of Milin's Theorem Ⅵ Borel Summability and General Circle Methods 1 Introduction 2 The Methods B and B' 3 Borel Summability of Power Series 4 The Borel Polygon 5 General Circle Methods Fλ 6 Auxiliary Estimates 7 Series with Ostrowski Gaps 8 Boundedness Results 9 Integral Formulas forLimitability 10 Integral Formulas: Case of Positive Sn 11 First Form of theTauberian Theorem 12 General Tauberian Theorem with Schmidt's Condition 13 Tauberian Theorem: Case of Positive Sn 14 AnApplication to Number Theory 15High-Indices Theorems 16 Restricted High-Indices Theorem for General Circle Methods 17 The Borel High-Indices Theorem 18 Discussion of the Tauberian Conditions 19 Growth of Power Series with Square-Root Gaps 20Euler Summability 21 The Taylor Method and Other Special Circle Methods 22 The Special Methods as Fλ-Methods 23High-Indices Theorems for Special Methods 24 Power Series Methods 25 Proof of Theorem24.4 Ⅶ Tauberian Remainder Theory 1 Introduction 2 Power Series and Laplace Transforms:How the Theory Developed 3 Theorems for Laplace Transforms 4 Proof of Theorems 3.1 and 3.2 5 One-Sided L 1 Approximation 6 Proof of Proposition 5.2 7 Approximation of Smooth Functions 8 Proof of Approximation Theorem 3.4 9 Vanishing Remainders: Theorem 3.3 10 Optimality of the Remainder Estimates 11 Dirichlet Series and High Indices 12 Proof of Theorem 11.2, Continued 13 The Fourier Integral Method: Introduction 14 Fourier Integral Method: A Model Theorem 15 Auxiliary Inequality of Ganelius 16 Proof of the Model Theorem 17 A More General Theorem 18 Application to Stieltjes Transforms 19 Fourier Integral Method: Laplace-Stieltjes Transform 20 Related Results 21 Nonlinear Problems of Erd6s for Sequences 22 Introduction to the Proof of Theorem 21.3 23 Proof of Theorem 21.3, Continued 24 An Example and Some Remarks 25 Introduction to the Proof of Theorem 21.5 26 The Fundamental Relation and a Reduction 27 Proof of Theorem 25.1, Continued 28 The End Game References Index
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