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不等式

版次：1
页数：
字数：
印刷时间：2012年03月01日
开本：12k
纸张：胶版纸
包装：平装
是否套装：否
国际标准书号ISBN：9787510042829

作者:(英)加林出版社:世界图书出版公司出版时间:2012年03月

内容简介

　　本书旨在介绍大量运用于线性分析中的不等式，并且详细介绍它们的具体应用。本书以柯西不等式开头，grothendieck不等式结束，中间用许多不等式串成一个完整的篇幅，如，loomiswhitney不等式、*值不等式、hardy 和 hilbert不等式、超收缩和拉格朗日索伯列夫不等、beckner以及等等。这些不等式可以用来研究函数空间的性质，它们之间的线性算子，以及*和算子。书中拥有许多完整和标准的结果，提供了许多应用，如勒贝格分解定理和勒贝格密度定理、希尔伯特变换和其他奇异积分算子、鞅收敛定理、特征值分布、lidskii积公式、mercer定理和littlewood 4/3定理。本书由(英)加林著。

目　　录

Introduction
1 Measure and integral
1.1 Measure
1.2 Measurable functions
1.3 Integration
1.4 Notes and remarks
2 The Cauchy-Schwarz inequality
2.1 Cauchy's inequality
2.2 Inner-product spaces
2.3 The Cauchy-Schwarz inequality
2.4 Notes and remarks
3 The AM-GM inequality
3.1 The AM-GM inequality
3.2 Applications

Introduction 
1 Measure and integral 
1.1 Measure 
1.2 Measurable functions 
1.3 Integration 
1.4 Notes and remarks 
2 The Cauchy-Schwarz inequality 
2.1 Cauchy's inequality 
2.2 Inner-product spaces 
2.3 The Cauchy-Schwarz inequality 
2.4 Notes and remarks 
3 The AM-GM inequality 
3.1 The AM-GM inequality 
3.2 Applications 
3.3 Notes and remarks 
4 Convexity and Jensen's inequality 
4.1 Convex sets and convex functions 
4.2 Convex functions on an interval 
4.3 Directional derivatives and sublinear functionals 
4.4 The Hahn-Banach theorem 
4.5 Normed spaces, Banach spaces and Hilbert space 
4.6 The Hahn-Banach theorem for normed spaces 
4.7 Barycentres and weak integrals 
4.8 Notes and remarks 
5 The Lp spaces 
5.1 Lp spaces, and Minkowski's inequality 
5.2 The Lebesgue decomposition theorem 
5.3 The reverse Minkowski inequality 
5.4 HSlder's inequality 
5.5 The inequalities of Liapounov and Littlewood 
5.6 Duality 
5.7 The Loomis-Whitney inequali'ty 
5.8 A Sobolev inequality 
5.9 Schur's theorem and Schur's test 
5.10 Hilbert's absolute inequality 
5.11 Notes and remarks 
6 Banach function spaces 
6.1 Banach function spaces 
6.2 Function space duality 
6.3 Orlicz space 
6.4 Notes and remarks 
7 Rearrangements 
7.1 Decreasing rearrangements 
7.2 Rearrangement-invariant Banach function spaces 
7.3 Muirhead's maximal function 
7.4 Majorization 
7.5 Calder6n's interpolation theorem and its converse 
7.6 Symmetric Banach sequence spaces 
7.7 The method of transference 
7.8 Finite doubly stochastic matrices 
7.9 Schur convexity 
7.10 Notes and remarks Maximal inequalities 
8.1 The Hardy-Riesz inequality 
8.2 The Hardy-Riesz inequality 
8.3 Related inequalities 
8.4 Strong type and weak type 
8.5 Riesz weak type 
8.6 Hardy, Littlewood, and a batsman's averages 
8.7 Riesz's sunrise lemma 
8.8 Differentiation almost everywhere 
8.9 Maximal operators in higher dimensions 
8.10 The Lebesgue density theorem 
8.11 Convolution kernels 
8.12 Hedberg's inequality 
8.13 Martingales 
8.14 Doob's inequality 
8.15 The martingale convergence theorem 
8.16 Notes and remarks 
9 Complex interpolation 
9.1 Hadamard's three lines inequality 
9.2 Compatible couples and intermediate spaces 
9.3 The Riesz-Thorin interpolation theorem 
9.4 Young's inequality 
9.5 The Hausdorff-Young inequality 
9.6 Fourier type 
9.7 The generalized Clarkson inequalities 
9.8 Uniform convexity 
9.9 Notes and remarks 
10 Real interpolation 
10.1 The Marcinkiewicz interpolation theorem: I 
10.2 Lorentz spaces 
10.3 Hardy's inequality 
10.4 The scale of Lorentz spaces 
10.5 The Marcinkiewicz interpolation theorem: II 
10.6 Notes and remarks 
11 The Hilbert transform, and Hilbert's inequalities 
11.1 The conjugate Poisson kernel 
11.2 The Hilbert transform on 
11.3 The Hilbert transform on 
11.4 Hilbert's inequality for sequences 
11.5 The Hilbert transform on T 
11.6 Multipliers 
11.7 Singular integral operators 
11.8 Singular integral operators on 
11.9 Notes and remarks 
12 Khintchine's inequality 
12.1 The contraction principle 
12.2 The reflection principle, and Lavy's inequalities 
12.3 Khintchine's inequality 
12.4 The law of the iterated logarithm 
12.5 Strongly embedded subspaces 
12.6 Stable random variables 
12.7 Sub-Gaussian random variables 
12.8 Kahane's theorem and Kahane's inequality 
12.9 Notes and remarks 
13 Hypercontractive and logarithmic Sobolev inequalities 
13.1 Bonami's inequality 
13.2 Kahane's inequality revisited 
13.3 The theorem of Lataa and Oleszkiewicz 
13.4 The logarithmic Sobolev inequality on Dd 
13.5 Gaussian measure and the Hermite polynomials 
13.6 The central limit theorem 
13.7 The Gaussian hypercontractive inequality 
13.8 Correlated Gaussian random variables 
13.9 The Gaussian logarithmic Sobolev inequality 
13.10 The logarithmic Sobolev inequality in higher dimensions 
13.11 Beckner's inequality 
13.12 The Babenko-Beckner inequality 
13.13 Notes and remarks 
14 Hadamard's inequality 
14.1 Hadamard's inequality 
14.2 Hadamard numbers 
14.3 Error-correcting codes 
14.4 Note and remark 
15 Hilbert space operator inequalities 
15.1 Jordan normal form 
15.2 Riesz operators 
15.3 Related operators 
15.4 Compact operators 
15.5 Positive compact operators 
15.6 Compact operators between Hilbert spaces 
15.7 Singular numbers, and the Rayleigh-Ritz minimax formula 
15.8 Weyl's inequality and Horn's inequality 
15.9 Ky Fan's inequality 
15.10 Operator ideals 
15.11 The Hilbert-Schmidt class 
15.12 The trace class 
15.13 Lidskii's trace formula 
15.14 Operator ideal duality 
15.15 Notes and remarks 
16 Summing operators 
16.1 Unconditional convergence 
16.2 Absolutely summing operators 
16.3 (p, q)-summing operators 
16.4 Examples of p-summing operators 
16.5 (p, 2)-summing operators between Hilbert spaces 
16.6 Positive operators on 
16.7 Mercer's theorem 
16.8 p-summing operators between Hilbert spaces 
16.9 Pietsch's domination theorem 
16.10 Pietsch's factorization theorem 
16.11 p-summing operators between Hilbert spaces 
16.12 The Dvoretzky-Rogers theorem 
16.13 Operators that factor through a Hilbert space 
16.14 Notes and remarks 
17 Approximation numbers and eigenvalues 
17.1 The approximation, Gelfand and Weyl numbers 
17.2 Subadditive and submultiplicative properties 
17.3 Pietsch's inequality 
17.4 Eigenvalues of p-summing and (p, 2)-summing
endomorphisms 
17.5 Notes and remarks 
18 Grothendieck's inequality, type and cotype 
18.1 Littlewood's 4/3 inequality 
18.2 Grothendieck's inequality 
18.3 Grothendieck's theorem 
18.4 Another proof, using Paley's inequality 
18.5 The little Grothendieck theorem 
18.6 Type and cotype 
18.7 Gaussian type and cotype 
18.8 Type and cotype of LP spaces 
18.9 The little Grothendieck theorem revisited 
18.10 More on cotype 
18.11 Notes and remarks 
References 
Index of inequalities 
Index

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