Schmüdgen K. The Moment Problem

New York: Springer, 2017. — 530 p.This advanced textbook provides a comprehensive and unified account of the moment problem. It covers the classical one-dimensional theory and its multidimensional generalization, including modern methods and recent developments.
In both the one-dimensional and multidimensional cases, the full and truncated moment problems are carefully treated separately. Fundamental concepts, results and methods are developed in detail and accompanied by numerous examples and exercises. Particular attention is given to powerful modern techniques such as real algebraic geometry and Hilbert space operators. A wide range of important aspects are covered, including the Nevanlinna parametrization for indeterminate moment problems, canonical and principal measures for truncated moment problems, the interplay between Positivstellensätze and moment problems on semi-algebraic sets, the fibre theorem, multidimensional determinacy theory, operator-theoretic approaches, and the existence theory and important special topics of multidimensional truncated moment problems.
The Moment Problem will be particularly useful to graduate students and researchers working on moment problems, functional analysis, complex analysis, harmonic analysis, real algebraic geometry, polynomial optimization, or systems theory. With notes providing useful background information and exercises of varying difficulty illustrating the theory, this book will also serve as a reference on the subject and can be used for self-study.Integral Representations of Linear Functionals
Moment Problems on Abelian ∗-SemigroupsOne-Dimensional Moment Problems on Intervals: Existence
One-Dimensional Moment Problems: Determinacy
Orthogonal Polynomials and Jacobi Operators
The Operator-Theoretic Approach to the Hamburger Moment Problem
The Indeterminate Hamburger Moment Problem
The Operator-Theoretic Approach to the Stieltjes Moment ProblemThe One-Dimensional Truncated Hamburger and Stieltjes Moment Problems
The One-Dimensional Truncated Moment Problem on a Bounded Interval
The Moment Problem on the Unit CircleThe Moment Problem on Compact Semi-Algebraic Sets
The Moment Problem on Closed Semi-Algebraic Sets: Existence
The Multidimensional Moment Problem: Determinacy
The Complex Moment Problem
Semidefinite Programming and Polynomial OptimizationMultidimensional Truncated Moment Problems: Existence
Multidimensional Truncated Moment Problems: Basic Concepts and Special Topics
The Truncated Moment Problem for Homogeneous Polynomials