陶伯理論

內(nèi)容概要

陶伯理論對級數(shù)和積分的可求和性判定的不同方法加以比較，確定它們何時收斂，給出漸近估計和余項(xiàng)估計。由陶伯理論的最初起源開始，作者介紹該理論的發(fā)展歷程：他的專業(yè)評論再現(xiàn)了早期結(jié)果所引來的興奮；論及困難而令人著迷的哈代-李特爾伍德定理及其出人意料的一個簡潔證明；高度贊揚(yáng)維納基于傅里葉理信論的突破，引人入勝的“高指數(shù)”定理以及應(yīng)用于概率論的Karamata正則變分理論。作者也提及蓋爾范德對維納理論的代數(shù)處理以及基本人的分布方法。介紹了博雷爾方法和“圓”方法的一個統(tǒng)一的新理論，本書還討論研究素數(shù)定理的各種陶伯方法。書后附有大量參考文獻(xiàn)和詳細(xì)盡的索引。

作者簡介

作者：(荷蘭)科雷瓦 (Jacob Korevaar)

書籍目錄

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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