陶伯理论

出版时间：2007-1  出版社：科学分社  作者：科雷瓦  页数：483  字数：594000  

内容概要

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。由陶伯理论的最初起源开始，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-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信论的突破，引人入胜的“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基本人的分布方法。介绍了博雷尔方法和“圆”方法的一个统一的新理论，本书还讨论研究素数定理的各种陶伯方法。书后附有大量参考文献和详细尽的索引。

作者简介

作者：(荷兰)科雷瓦 (Jacob Korevaar)

书籍目录

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 FunctionⅡ 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 FunctionsⅢ 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 TransformⅣ 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 TransformsⅤ 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 TheoremⅥ 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 Theorem24.4Ⅶ 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 GameReferencesIndex

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《国外数学名著系列(影印版)36:陶伯理论百年进展》是科学出版社出版。

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