陶伯理论对级数和积分的可求和性判定的不同方法加以比较，确定它们何时收敛，给出渐近估计和余项估计。由陶伯理论的*初起源开始，作者介绍该理论的发展历程：他的专业评论再现了早期结果所引来的兴奋；论及困难而令人着迷的哈代一李特尔伍德定理及其出人意料的一个简洁证明；高度赞扬维纳基于傅里叶理论的突破，引人人胜的“高指数”定理以及应用于概率论的Karamata正则变分理论。作者也提及盖尔范德对维纳理论的代数处理以及其本人的分布方法。介绍了博雷尔方法和“圆”方法的一个统一的新理论，《国外数学名著系列（影印版）36：陶伯理论 百年进展》还讨论了研究素定理的各种陶伯方法。书后附有大量参考文献和详尽的索引。

Ⅰ 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

Ⅱ 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

Ⅴ 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
15 High-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
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 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