半群上的调和分析
2012-8
世界图书出版公司
博格
289
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The purpose of this book is to provide a treatment of these
positive definite functions on abelian semigroups with involution.
In doing so we also discuss related topics such as negative
definite functions， completely monotone functions and
Hoeffding-type inequalities. We view these subjects as important
ingredients of harmonic analysis on semigroups. It has been our
aim， simultaneously， to write a book which can serve as a textbook
for an advanced graduate course， because we feel that the notion of
positive definiteness is an important and basic notion which occurs
in mathematics as often as the notion of a Hilbert space. The
already mentioned Laplace and ourier transformations， as well as
the generating functions for integer valued random variables，
belong to the most important analytical tools in probability theory
and its applications. Only recently it turned out that positive
（resp. negative） definite functions allow a probabilistic
characterization in terms of so-called Hoeffding-type
inequalities.
作者：（丹麦）博格（Christian Berg）
CHAPTER 1 Introduction to Locally Convex Topological Vector
Spaces and Dual Pairs
1.Locally Convex Vector Spaces
2.Hahn-Banach Theorems
3.Dual Pairs
Notes and Remarks
CHAPTER 2 Radon Measures and Integral Representations
1.Introduction to Radon Measures on Hausdorff Spaces
2.The Riesz Representation Theorem
3.Weak Convergence of Finite Radon Measures
4.Vague Convergence of Radon Measures on Locally Compact
Spaces
5.Introduction to the Theory of lntegral Representations
Notes and Remarks
CHAPTER 3 General Results on Positive and Negative Definite
Matrices and Kernels
1.Definitions and Some Simple Properties of Positive and Negative
Definite Kernels
2.Relations Between Positive and Negative Definite Kernels
3.Hilbert Space Representation of Positive and Negative Definite
Kernels
Notes and Remarks
CHAPTER 4 Main Results on Positive and Negative Definite Functions
on Semigroups
1.Definitions and Simple Properties
2.Exponentially Bounded Positive Definite Functions on Abelian
Semigroups
3.Negative Definite Functions on Abelian Semigroups
4.Examples of Positive and Negative Defnite Functions
5.Positive Functions
6.Completely Monotone and Alternating Functions
Notes and Remarks
CHAPTER 5 Schoenberg-Type Results for Positive and Negative
Definite Functions
1.Schoenberg Triples
2.Norm Dependent Positive Definite Functions on Banach Spaces
3.Functions Operating on Positive Definite Matrices
4.Schoenberg's Theorem for the Complex Hilbert Sphere
5.The Real Infinite Dimensional Hypcrbolic Space
Notes and Remarks
CHAPTER 6 Positive Definite Functions and Moment Functions
1.Moment Functions
2.The One-Dimensional Moment Problem
3.The Multi Dimensional Moment Problem
4.The Two-Sided Moment Problem
5.Perfect Semigroups
Notes and Remarks
CHAPTER 7 Hoeffding's Inequality and Multivariate
Majorization
1.The Discrete Case
2.Extension to Nondiscrete Semigroups
3.Completely Negative Definite Functions and Schur-Monotonicity
Notes and Remarks
CHAPTER 8 Positive and Negative Definite Functions on Abelian
Semigroups Without Zero
1.Quasibounded Positive and Negative Definite Functions
2.Completely Monotone and Completely Alternating Functions
Notes and Remarks
References
List of Symbols
Index
版权页： 插图： Notes and Remarks In his fundamental paper "Recherches sur les fractions continues", Stieltjesformulated and solved the moment problem which bears his name. LaterHamburger generalized Stieitjes' result to moment sequences of measureson the whole real line. Moments of a measure have been studied, beforeStieltjes, by Tchebycheff and others, but concerning the history of themoment problem we refer to Shohat and Tamarkin (1943). Our choice of thename "moment function" on a semigroup S is motivated by this classicaltheory, which corresponds to the semigroup (No, +). Likewise the symbol(S) reflects the name of Hamburger. The results in 1.6-1.11 seem to be newbut are, of course,known for some concrete semigroups. For a detailed study of Hamburger's moment problem, in particular ofthe set E+(R, s), we refer the reader to the classical monographs by Akhiezer(1965) and Shohat and Tamarkin (1943). Results about denseness of the setof polynomials in P(R,μ) can be found in Berg and Christensen (1981,1983a). The F-moment problem in the case F = {x ∈ R|p(x) ≥ 0}, where pis a fixed polynomial, is studied in Berg and Maserick (1982). It contains acharacterization of the polynomials p for which the set of {p≥0}-momentsequences is equal to {s|s, p(E)s e(No)}. HereThe F-moment problem where F = R\Uni=1]ai, bi[ and a1 < b1 < a2
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