不等式

內(nèi)容概要

　　本書旨在介紹大量運(yùn)用于線性分析中的不等式，并且詳細(xì)介紹它們的具體應(yīng)用。本書以柯西不等式開頭，grothendieck不等式結(jié)束，中間用許多不等式串成一個(gè)完整的篇幅，如，loomiswhitney不等式、最大值不等式、hardy
和
hilbert不等式、超收縮和拉格朗日索伯列夫不等、beckner以及等等。這些不等式可以用來研究函數(shù)空間的性質(zhì)，它們之間的線性算子，以及絕對(duì)和算子。書中擁有許多完整和標(biāo)準(zhǔn)的結(jié)果，提供了許多應(yīng)用，如勒貝格分解定理和勒貝格密度定理、希爾伯特變換和其他奇異積分算子、鞅收斂定理、特征值分布、lidskii積公式、mercer定理和littlewood
4/3定理。本書由(英)加林著。

作者簡(jiǎn)介

作者:(英)加林

書籍目錄

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

章節(jié)摘錄

版權(quán)頁：   插圖：   Many of the inequalities that we shall establish originally concern finitesequences and finite sums． We then extend them to infinite sequences andinfinite sums， and to functions and integrals， and it is these more generalresults that are useful in applications． Although the applications can be useful in simple settings—concerning the Riemann integral of a continuous function， for example—the extensions areusually made by a limiting process． For this reason we need to work in themore general setting of measure theory， where appropriate limit theoremshold． We give a brief account of what we need to know； the details of the theory will not be needed， although it is hoped that the results that weeventually establish will encourage the reader to master them． If you arenot familiar with measure theory， read through this chapter quickly， and then come back to it when you find that the need arises． Suppose that Ω is a set． A measure ascribes a size to some of the subsetsof Ω． It turns out that we usually cannot do this in a sensible way for all the subsets of Ω， and have to restrict attention to the measurable subsets of Ω． These are the 'good' subsets of Ω， and include all the sets that we meet in practice． The collection of measurable sets has a rich enough structure that we can carry out countable limiting operations． A σ—field ∑ is a collection of subsets of a set Ω which satisfies （i） if （Ai） is a sequence in ∑ then Ui∞=1Ai ∈ ∑， and （ii） if A ∈ ∑ then the complement Ω \ A ∈ ∑． Thus （iii） if （Ai） is a sequence in ∑ then ∩i∞=1Ai ∈ ∑． The sets in ∑ are called ∑—measurable sets； if it is clear what ∑ is， they are simply called the measurable sets．

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