陶伯理论:百年进展:A century of developments

Ⅰ The Hardy-Littlewood Theorems
1 Introduction
2 Examples of Summability Methods AbeliaTheorems and TauberiaQuestion
3 Simple Applications of Cesa(‘)ro， Abel and Borel Summability
4 Lambert Summability iNumber Theory
5 Tauber’s Theorems for Abel Summability
6 TauberiaTheorem for Cesa(‘)ro Summability
7 Hardy-Littlewood Tauberians for Abel Summability
8 Tauberians Involving Dirichlet Series
9 Tauberians for Borel Summability
10 Lambert Tauberiaand Prime Number Theorem
11 Karamata’s Method for Power Series
12 Wielandt’s Variatioothe Method
13 Transitiofrom Series to Integrals
14 Extensioof Tauber’s Theorems to Laplace-Stieltjes Transforms
15 Hardy-Littlewood Type Theorems Involving Laplace Transforms
16 Other TauberiaConditions： 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 TauberiaConditions
25 TauberiaTheorems of Nonstandard Type
26 Important Properties of the Zeta Function

Ⅱ Wiener’s Theory
1 Introduction
2 Wiener Problem： Pitt’s Form
3 Testing Equatiofor 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 DivisioTheorem
10 Wiener Families of Kernels
11 Distributional Approach to Wiener Theory
12 General Tauberiafor Lambert SummabilitY
13 Wiener’s ‘Second TauberiaTheorem’
14 A Wiener Theorem for Series
15 Extensions
16 Discussioof the TauberiaConditions
17 Landau-Ingham Asymptotics
18 Ingham Summability
19 Applicatioof Wiener Theory to Harmonic Functions

Ⅲ Complex TauberiaTheorems
1 Introduction
2 A Landau-Type Tauberiafor Dirichlet Series
3 MelliTransforms
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 Applicatioto Dirichlet Series and the PNT
9 Laplace Transforms of Functions Bounded From Below
10 TauberiaConditions Other ThaBoundedness
11 AOptimal Constant iTheorem 10．1
12 Fatou and Riesz． General Dirichlet Series
13 Newer Extensions of Fatou-Riesz
14 PseudofunctioBoundary Behavior
15 Applications to Operator Theory
16 Complex Remainder Theory
17 The Remainder iFatou’s Theorem
18 Remainders iHardy-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 Characterizatioof 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 iHigher-Order Theory
13 MerceriaTheorems
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 Applicatioto Exponential Growth
19 Very Large Laplace Transforms
20 Logarithmic Theory for Very Large Transforms
21 Large Transforms： Complex Approach
22 Proof of Propositio21．4
23 Asymptotics for Partitions
24 Two-Sided Laplace Transforms

Ⅴ Extensions of the Classical Theory
1 Introduction
2 Preliminaries oBanach Algebras
3 Algebraic Form of Wiener’s Theorem
4 Weighted L1 Spaces
5 Gelfand’s Theory of Maximal Ideals
6 Applicatioto the Banach Algebra Aω = (Lω， C)
7 Regularity Conditiofor Lω
8 The Closed Maximal Ideals iLω
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 aFK-space
16 Perfect Matrix Methods
17 Methods with Sectional Convergence
18 Existence of (Limitable) Bounded Divergent Sequences
19 Bounded Divergent Sequences， Continued
20 Gap TauberiaTheorems
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 theTauberiaTheorem
12 General TauberiaTheorem with Schmidt’s Condition
13 TauberiaTheorem： Case of Positive Sn
14 AnApplicatioto Number Theory
15 High-Indices Theorems
16 Restricted High-Indices Theorem for General Circle Methods
17 The Borel High-Indices Theorem
18 Discussioof the TauberiaConditions
19 Growth of Power Series with Square-Root Gaps
20 Euler Summability
21 The Taylor Method and Other Special Circle Methods
22 The Special Methods as Fλ-Methods
23 High-Indices Theorems for Special Methods
24 Power Series Methods
25 Proof of Theorem24．4

Ⅶ TauberiaRemainder 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 Propositio5．2
7 Approximatioof Smooth Functions
8 Proof of ApproximatioTheorem 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 Applicatioto Stieltjes Transforms
19 Fourier Integral Method： Laplace-Stieltjes Transform
20 Related Results
21 Nonlinear Problems of Erd6s for Sequences
22 Introductioto the Proof of Theorem 21．3
23 Proof of Theorem 21．3， Continued
24 AExample and Some Remarks
25 Introductioto the Proof of Theorem 21．5
26 The Fundamental Relatioand a Reduction
27 Proof of Theorem 25．1， Continued
28 The End Game
References
Index