The Moment Problem / by Konrad Schmüdgen.

Format:

Book

Author/Creator:

Schmüdgen, Konrad., Author.

Series:

Graduate Texts in Mathematics, 2197-5612 ; 277

Language:

English

Subjects (All):

Functional analysis.

Operator theory.

Algebraic fields.

Polynomials.

Geometry, Algebraic.

Functional Analysis.

Operator Theory.

Field Theory and Polynomials.

Algebraic Geometry.

Local Subjects:

Functional Analysis.

Operator Theory.

Field Theory and Polynomials.

Algebraic Geometry.

Physical Description:

1 online resource (XXIII, 512 p. 6 illus.)

Edition:

1st ed. 2017.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2017.

Summary:

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.

Contents:

Preface and Overview

1 Integral representations of linear functionals

2 Moment problems on abelian *-semigroups

Part I The one dimensional moment problem

3 One dimensional moment problems on intervals: existence

4 One dimensional moment problems: determinacy

5 Orthogonal polynomials and Jacobi operators

6 The operator-theoretic approach to the Hamburger moment problem

7 The indeterminate Hamburger moment problem

8 The operator-theoretic approach to the Stieltjes moment problem

Part II The one dimensional truncated moment problem

9 The one-dimensional truncated Hamburger and Stieltjes moment problems

10 The one-dimensional truncated moment problem on a bounded interval

11 The moment problem on the unit circle

Part III The multidimensional moment problem

12 The moment problem on compact semi-algebraic sets

13 The moment problem on closed semi-algebraic sets: existence

14 The multidimensional moment problem: determinacy

15 The complex moment problem

16 Semidefinite programming and polynomial optimization

Part IV The multidimensional truncated moment problem

17 Multidimensional truncated moment problems: existence criteria

18 Multidimensional truncated moment problems: basic concepts and special topics

19 The truncated moment problem for homogeneous polynomials.

Notes:

Includes bibliographical references and index.

ISBN:

3-319-64546-3

OCLC:

1012401036