Graduate Texts in Mathematics 277

Graduate Texts in Mathematics

Series Editors:

Sheldon AxlerSan Francisco State University, San Francisco, CA, USA

Kenneth RibetUniversity of California, Berkeley, CA, USA

Advisory Board:

Alejandro Adem, University of British ColumbiaDavid Eisenbud, University of California, Berkeley & MSRIIrene M. Gamba, The University of Texas at AustinJ.F. Jardine, University of Western OntarioJeffrey C. Lagarias, University of MichiganKen Ono, Emory UniversityJeremy Quastel, University of TorontoFadil Santosa, University of MinnesotaBarry Simon, California Institute of Technology

Graduate Texts in Mathematics bridge the gap between passive study andcreative understanding, offering graduate-level introductions to advanced topicsin mathematics. The volumes are carefully written as teaching aids and highlightcharacteristic features of the theory. Although these books are frequently used astextbooks in graduate courses, they are also suitable for individual study.

More information about this series at http://www.springer.com/series/136

Konrad Schmüdgen

The Moment Problem

123

Konrad SchmüdgenMathematisches InstitutUniversität LeipzigGermany

ISSN 0072-5285 ISSN 2197-5612 (electronic)Graduate Texts in MathematicsISBN 978-3-319-64545-2 ISBN 978-3-319-64546-9 (eBook)DOI 10.1007/978-3-319-64546-9

Library of Congress Control Number: 2017951141

Mathematics Subject Classification (2010): 44A60, 14P10, 47A57

© Springer International Publishing AG 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To KATJA

Thomas Jean Stieltjes (1856 – 1894)

Pafnuty Lvovich Chebyshev Andrey Andreyevich Markov(1821 – 1894) (1856 – 1922)

Contents

1 Integral Representations of Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . 131.1 Integral Representations of Functionals on Adapted Spaces . . . . . 14

1.1.1 Moment Functionals and Adapted Spaces. . . . . . . . . . . . . . . . . 141.1.2 Existence of Integral Representations . . . . . . . . . . . . . . . . . . . . . 151.1.3 The Set of Representing Measures. . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Integral Representations of Functionals onFinite-Dimensional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.1 Atomic Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.2.2 Strictly Positive Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . 251.2.3 Sets of Atoms and Determinate Moment Functionals . . . . 271.2.4 Supporting Hyperplanes of the Cone of Moment

Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2.5 The Set of Atoms and the Core Variety . . . . . . . . . . . . . . . . . . . 341.2.6 Extremal Values of Integrals with Moment

Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.4 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2 Moment Problems on Abelian �-Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.1 �-Algebras and �-Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.2 Commutative �-Algebras and Abelian �-Semigroups . . . . . . . . . . . . 462.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.1 Example 1: Nd0; n� D n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.2 Example 2: N2d0 ; .n;m/� D .m; n/ . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.3.3 Example 3: Zd; n� D �n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3.4 Example 4: Z; n� D n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.5 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

vii

viii Contents

Part I The One-Dimensional Moment Problem

3 One-Dimensional Moment Problems on Intervals: Existence . . . . . . . . . 573.1 Positive Polynomials on Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 The Moment Problem on Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 The Symmetric Hamburger Moment Problem and Stieltjes

Moment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4 Positive Polynomials on Intervals (Revisited) . . . . . . . . . . . . . . . . . . . . . 693.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 One-Dimensional Moment Problems: Determinacy . . . . . . . . . . . . . . . . . . . . 794.1 Measures Supported on Bounded Intervals . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Carleman’s Condition.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3 Krein’s Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.5 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 Orthogonal Polynomials and Jacobi Operators . . . . . . . . . . . . . . . . . . . . . . . . . 935.1 Definitions of Orthogonal Polynomials and Explicit Formulas. . . 945.2 Three Term Recurrence Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.3 The Moment Problem and Jacobi Operators . . . . . . . . . . . . . . . . . . . . . . 1025.4 Polynomials of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.5 The Wronskian and Some Useful Identities . . . . . . . . . . . . . . . . . . . . . . . 1085.6 Zeros of Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.7 Symmetric Moment Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6 The Operator-Theoretic Approach to the Hamburger MomentProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.1 Existence of Solutions of the Hamburger Moment Problem.. . . . . 1216.2 The Adjoint of the Jacobi Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.3 Determinacy of the Hamburger Moment Problem . . . . . . . . . . . . . . . . 1256.4 Determinacy Criteria Based on the Jacobi Operator . . . . . . . . . . . . . . 1296.5 Self-Adjoint Extensions of the Jacobi Operator .. . . . . . . . . . . . . . . . . . 1336.6 Markov’s Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.7 Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7 The Indeterminate Hamburger Moment Problem . . . . . . . . . . . . . . . . . . . . . . 1457.1 The Nevanlinna Functions A.z/;B.z/;C.z/;D.z/ . . . . . . . . . . . . . . . . . 1457.2 Von Neumann Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.3 Weyl Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.4 Nevanlinna Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.5 Maximal Point Masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Contents ix

7.6 Nevanlinna–Pick Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.7 Solutions of Finite Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1737.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8 The Operator-Theoretic Approach to the Stieltjes MomentProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.1 Preliminaries on Quadratic Forms on Hilbert Spaces . . . . . . . . . . . . . 1778.2 Existence of Solutions of the Stieltjes Moment Problem . . . . . . . . . 1798.3 Determinacy of the Stieltjes Moment Problem .. . . . . . . . . . . . . . . . . . . 1808.4 Friedrichs and Krein Approximants.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1848.5 Nevanlinna Parametrization for the Indeterminate Stieltjes

Moment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.6 Weyl Circles for the Indeterminate Stieltjes Moment Problem .. . 1968.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1998.8 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Part II The One-Dimensional Truncated Moment Problem

9 The One-Dimensional Truncated Hamburger and StieltjesMoment Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2039.1 Quadrature Formulas and the Truncated Moment Problem

for Positive Definite 2n-Sequences.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2049.2 Hamburger’s Theorem and Markov’s Theorem Revisited . . . . . . . . 2099.3 The Reproducing Kernel and the Christoffel Function.. . . . . . . . . . . 2119.4 Positive Semidefinite 2n-Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2149.5 The Hankel Rank of a Positive Semidefinite 2n-Sequence . . . . . . . 2179.6 Truncated Hamburger and Stieltjes Moment Sequences .. . . . . . . . . 2219.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2289.8 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

10 The One-Dimensional Truncated Moment Problem on aBounded Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22910.1 Existence of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22910.2 The Moment Cone SmC1 and Its Boundary Points . . . . . . . . . . . . . . . . 23110.3 Interior Points of SmC1 and Interlacing Properties of Roots . . . . . . 23410.4 Principal Measures of Interior Points of SmC1 . . . . . . . . . . . . . . . . . . . . 23710.5 Maximal Masses and Canonical Measures . . . . . . . . . . . . . . . . . . . . . . . . 24110.6 A Parametrization of Canonical Measures. . . . . . . . . . . . . . . . . . . . . . . . . 24610.7 Orthogonal Polynomials and Maximal Masses. . . . . . . . . . . . . . . . . . . . 24810.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25410.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

11 The Moment Problem on the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25711.1 The Fejér–Riesz Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25711.2 Trigonometric Moment Problem: Existence of a Solution .. . . . . . . 25911.3 Orthogonal Polynomials on the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . 261

x Contents

11.4 The Truncated Trigonometric Moment Problem . . . . . . . . . . . . . . . . . . 26611.5 Carathéodory Functions, the Schur Algorithm,

and Geromimus’ Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27111.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27811.7 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Part III The Multidimensional Moment Problem

12 The Moment Problem on Compact Semi-Algebraic Sets . . . . . . . . . . . . . . 28312.1 Semi-Algebraic Sets and Positivstellensätze . . . . . . . . . . . . . . . . . . . . . . 28412.2 Localizing Functionals and Supports of Representing

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28812.3 The Moment Problem on Compact Semi-Algebraic Sets

and the Strict Positivstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29312.4 The Representation Theorem for Archimedean Modules . . . . . . . . . 29812.5 The Operator-Theoretic Approach to the Moment Problem .. . . . . 30212.6 The Moment Problem for Semi-Algebraic Sets Contained

in Compact Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30712.7 Examples and Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30812.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31112.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

13 The Moment Problem on Closed Semi-Algebraic Sets: Existence . . . . 31513.1 Positive Polynomials and Sums of Squares. . . . . . . . . . . . . . . . . . . . . . . . 31613.2 Properties (MP) and (SMP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32013.3 The Fibre Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32213.4 (SMP) for Basic Closed Semi-Algebraic Subsets

of the Real Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32613.5 Application of the Fibre Theorem: Cylinder Sets

with Compact Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33013.6 Application of the Fibre Theorem: The Rational Moment

Problem on Rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33213.7 Application of the Fibre Theorem: A Characterization

of Moment Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33513.8 Closedness and Stability of Quadratic Modules . . . . . . . . . . . . . . . . . . . 33813.9 The Moment Problem on Some Cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34413.10 Proofs of the Main Implications of Theorems 13.10

and 13.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34813.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35313.12 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

14 The Multidimensional Moment Problem: Determinacy . . . . . . . . . . . . . . . 35714.1 Various Notions of Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35714.2 Polynomial Approximation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36114.3 Partial Determinacy and Moment Functionals . . . . . . . . . . . . . . . . . . . . 36414.4 The Multivariate Carleman Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

Contents xi

14.5 Moments of Gaussian Measure and Surface Measureon the Unit Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

14.6 Disintegration Techniques and Determinacy . . . . . . . . . . . . . . . . . . . . . . 37414.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37814.8 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

15 The Complex Moment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38115.1 Relations Between Complex and Real Moment Problems.. . . . . . . 38115.2 The Moment Problems for the �-Semigroups

Zd and N0 � Zd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38415.3 The Operator-Theoretic Approach to the Complex

Moment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38615.4 The Complex Carleman Condition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39115.5 An Extension Theorem for the Complex Moment Problem . . . . . . 39315.6 The Two-Sided Complex Moment Problem .. . . . . . . . . . . . . . . . . . . . . . 39515.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39715.8 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

16 Semidefinite Programming and Polynomial Optimization. . . . . . . . . . . . . 39916.1 Semidefinite Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39916.2 Lasserre Relaxations of Polynomial Optimization with

Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40316.3 Polynomial Optimization with Constraints . . . . . . . . . . . . . . . . . . . . . . . . 40516.4 Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40816.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41016.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

Part IV The Multidimensional Truncated Moment Problem

17 Multidimensional Truncated Moment Problems: Existence . . . . . . . . . . . 41517.1 The Truncated K-Moment Problem and Existence . . . . . . . . . . . . . . . 41617.2 The Truncated Moment Problem on Projective Space . . . . . . . . . . . . 42017.3 Hankel Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42517.4 Hankel Matrices of Functionals with Finitely Atomic

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42917.5 The Full Moment Problem with Finite Rank Hankel Matrix . . . . . 43317.6 Flat Extensions and the Flat Extension Theorem.. . . . . . . . . . . . . . . . . 43417.7 Proof of Theorem 17.36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43817.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44117.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

18 Multidimensional Truncated Moment Problems: BasicConcepts and Special Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44518.1 The Cone of Truncated Moment Functionals. . . . . . . . . . . . . . . . . . . . . . 44518.2 Inner Moment Functionals and Boundary

Moment Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45118.3 The Core Variety of a Linear Functional. . . . . . . . . . . . . . . . . . . . . . . . . . . 453

xii Contents

18.4 Maximal Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45618.5 Constructing Ordered Maximal Mass Measures . . . . . . . . . . . . . . . . . . 46218.6 Evaluation Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46418.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46918.8 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

19 The Truncated Moment Problem for Homogeneous Polynomials . . . . 47119.1 The Apolar Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47119.2 The Apolar Scalar Product and Differential Operators. . . . . . . . . . . . 47519.3 The Apolar Scalar Product and the Truncated

Moment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47719.4 Robinson’s Polynomial and Some Examples . . . . . . . . . . . . . . . . . . . . . . 48319.5 Zeros of Positive Homogeneous Polynomials .. . . . . . . . . . . . . . . . . . . . 48719.6 Applications to the Truncated Moment Problem on R2 . . . . . . . . . . 49119.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49619.8 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499A.1 Measure Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499A.2 Pick Functions and Stieltjes Transforms .. . . . . . . . . . . . . . . . . . . . . . . . . . 502A.3 Positive Semidefinite and Positive Definite Matrices . . . . . . . . . . . . . 504A.4 Positive Semidefinite Block Matrices and Flat Extensions . . . . . . . 506A.5 Locally Convex Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509A.6 Convex Sets and Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510A.7 Symmetric and Self-Adjoint Operators on Hilbert Space . . . . . . . . . 514

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Preface and Overview

Let � be a positive Radon measure on a closed subset K of Rd and ˛ D .˛1; : : : ; ˛d/

a multi-index of nonnegative integers ˛j. If the integral

s˛.�/ WDZKx˛1

1 : : : x˛dd d�.x/

is finite, the number s˛ � s˛.�/ is called the ˛-th moment of the measure �. If all˛-th moments exist, the sequence .s˛/˛2Nd

0is called the moment sequence of �.

Moments appear at many places in physics and mathematics. For instance, if Kis a solid body in R3 with mass density m.x1; x2; x3/, then the number

s.0;2;0/ C s.0;0;2/ DZK

.x22 C x2

3/m.x1; x2; x3/dx1dx2dx3

is the moment of interia of the body with respect to the x1-axis. Or if X is a randomvariable with distribution function F.x/, the expectation value of Xk is defined by

EŒXk� D sk DZ C1

�1xkdF.x/

and the variance of X is Var.X/ D EŒ.X � EŒX�/2� D EŒX2� � EŒX�2 D s2 � s21

(provided that these numbers are finite).The moment problem is a classical mathematical problem. In its simplest form,

the Hamburger moment problem for the real line, it is the following question:Let s D .sn/n2N0 be a real sequence. Does there exist a positive Radon measure

� on R such that for all n 2 N0 the integralR C1

�1 xnd� converges and satisfies

sn DZ C1

�1xn d� ‹ (1)

1

2 Preface and Overview

That is, the moment problem is the inverse problem of “finding” a representingmeasure � when the moment sequence s is given.

For a real sequence s D .sn/n2N0 let Ls denote the linear functional on thepolynomial algebra RŒx� (or on CŒx�) defined by Ls.xn/ D sn, n 2 N0. By thelinearity of the integral it is clear that (1) holds for all n 2 N0 if and only if we have

Ls.p/ DZ C1

�1p.x/ d� for p 2 RŒx�: (2)

Thus, the moment problem asks whether or not the functional Ls on RŒx� admitsan integral representation (2) with respect to some positive measure �. To getsome flavour of what this book is about, we sketch without giving proofs somecornerstones from the theory of the one-dimensional Hamburger problem.

Assume that s D .sn/n2N0 is the moment sequence of some positive measure �

on R. Then, for any polynomial p.x/ D PnkD0 akx

k 2 RŒx� we obtain

Ls.p2/ D

Zp.x/2 d� D

Z � nXk;lD0

akalxkCl

�d� D

nXk;lD0

akalskCl � 0:

Therefore, Ls.p2/ � 0 for all p 2 RŒx�, that is, the functional Ls is positive, and theHankel matrix Hn.s/ WD .skCl/

nk;lD0 is positive semidefinite for each n 2 N0. These

are two (equivalent) necessary conditions for a sequence to be a moment sequence.Hamburger’s theorem (1920) says that each of these necessary conditions is alsosufficient for the existence of a positive measure. That is, the existence problem fora solution is easily answered in terms of positivity conditions.

The question concerning the uniqueness of representing measures is more subtle.A moment sequence is called determinate if it has only one representing measure.For instance, the lognormal distribution

F.x/ D 1p2�

�.0;C1/.x/x�1e�.log x/2=2

gives a probability measure whose moment sequence is not determinate.Let us assume that s has a representing measure � supported on the bounded

interval Œ�a; a�, a > 0. It is not difficult to show that the support of any otherrepresenting measure, say Q�, is also contained in Œ�a; a�. Then (2) implies that

Z a

�af .x/d� D

Z a

�af .x/ d Q� (3)

for f 2 RŒx�. By Weierstrass’ theorem each continuous functions on Œ�a; a� canbe approximated uniformly by polynomials. Therefore (3) holds for all continuousfunctions f on Œ�a; a�. Hence � D Q�, so that s is determinate.

Preface and Overview 3

A useful sufficient criterion for determinacy is the Carleman condition

1XnD1

s� 1

2n2n D C1:

Now suppose s is a moment sequence such that Ls.pp/ > 0 for p 2 CŒx�; p ¤ 0,or equivalently, s has a representing measure with infinite support. Then

hp; qis D Ls.pq/; p; q 2 CŒx�;

defines a scalar product on the vector space CŒx�. Let X denote the multiplicationoperator by the variable x on CŒx�, that is, .Xp/.x/ D xp.x/. Applying the Gram–Schmidt procedure to the sequence .xn/n2N0 of the unitary space .CŒx�; h�; �is/ yieldsan orthonormal sequence .pn/n2N0 of polynomials pn 2 CŒx� such that each pn hasdegree n and a positive leading coefficient. Then there exist numbers an > 0 andbn 2 R such that the operator X acts on the orthonormal base .pn/ by

Xpn.x/ D anpnC1.x/ C bnpn.x/ C an�1pn�1.x/; n 2 N0; where p�1 WD 0:

That is, X is unitarily equivalent to the Jacobi operator TJ for the Jacobi matrix

J D

0BBBBB@

b0 a0 0 0 : : :

a0 b1 a1 0 : : :

0 a1 b2 a2 : : :

0 0 a2 b3 : : :

: : :: : :

: : :: : :

1CCCCCA

If A is a self-adjoint extension of TJ on a possibly larger Hilbert space and EA isthe spectral measure of A, then � D s0hEA.�/1; 1i is a solution of the momentproblem for s. Each solution is of this form. Further, the moment sequence s isdeterminate if and only if the closure of the symmetric operator X on the Hilbertspace completion of .CŒx�; h�; �is/ is self-adjoint. All these facts relate the momentproblem to the theory of orthogonal polynomials and to operator theory.

Let s be an indeterminate moment sequence. Nevanlinna’s theorem yields aparametrization of the solution set. Let P denote the set of holomorphic functionson the upper half-plane with nonnegative imaginary part and set P WD P [ f1g.Then Nevanlinna’s theorem states that there is a bijection � ! �� of P to the setof all solutions of the moment problems for s given by

Z 1

�11

x � zd��.x/ D �A.z/ C �.z/C.z/

B.z/ C �.z/D.z/; z 2 C; Im z > 0:

4 Preface and Overview

Here A;B;C;D are certain entire functions depending only on the sequence s. Thus,a moment sequence is either determinate or it has a “huge” family of representingmeasures. All this and much more is developed in detail in Part I of the book.

There are a number of other variants of the moment problem. The Stieltjesmoment problem asks for representing measures on the half-line Œ0; C1/ and theHausdorff moment problem for measures on the interval Œ0; 1�. If only the firstmoments s0; : : : ; sm are prescribed, we have a truncated moment problem. In thiscase there exist finitely atomic representing measures and one is interested inmeasures with “small” numbers of atoms.

The passage from one-dimensional to multidimensional moment problems leadsto fundamental new difficulties. As already observed by Hilbert, there are positivepolynomials in d � 2 variables that are not sums of squares. As a consequence,there exists a linear functional L on RdŒx� � RŒx1; : : : ; xd� which is positive (thatis, L.p2/ � 0 for p 2 RdŒx�), but L is not a moment functional (that is, it cannot bewritten in the form (2)). For this reason the K-moment problem is invented. It asksfor representing measures with support contained in a given closed subset K of Rd.

Let ff1; : : : ; fkg be a finite subset of RdŒx�. Then the closed set

K D fx 2 Rd W f1.x/ � 0; : : : ; fk.x/ � 0g

is called a semi-algebraic set. For such sets methods from real algebraic geometryprovide powerful tools for the study of the K-moment problem. Our main result fora compact semi-algebraic set K is the following: A linear functional L on RdŒx� is amoment functional with representing measure supported on K if and only if

L.f e1

1 � � � f ekk p2/ � 0 for p 2 RdŒx�; e1; : : : ; ek 2 f0; 1g:

If K is only closed but not compact, this is no longer true. But if there exist nontrivialbounded polynomials on K, there is a fibre theorem that allows one to reduce the K-moment problem to “lower dimensional” cases. The multidimensional determinacyquestion is much more complicated than its one-dimensional counterpart.

It turns out that most methods that have been successfully applied to the momentproblem in dimension one either fail in higher dimensions or at least require moreinvolved additional technical considerations.

The study of moment problems and related topics goes back to the late nineteenthcentury. The Russian mathematicians P.L. Chebychev (1974) and A.A. Markov(1984) applied moments in their “theory of limiting values of integrals” and inventedimportant notions; a survey of their ideas and further developments was givenby M.G. Krein [Kr2]. The moment problem itself as a problem in its own wasformulated for the first time by the Dutch mathematician T.J. Stieltjes (1894) inhis pioneering memoir [Stj]. Stieltjes treated this problem for measures supportedon the half-line and developed a far reaching theory. The cases of the real line and ofbounded intervals were studied only later by H. Hamburger (1920) and F. Hausdorff

Preface and Overview 5

(1920). Important early contributions have been made by R. Nevanlinna, M. Riesz,T. Carleman, M.H. Stone, and others.

A surprising feature of the moment problem theory is the connections andthe close interplay with many branches of mathematics and the broad range ofapplications. H.J. Landau wrote in the introduction of an article in the AMS volume“Moments in Mathematics” [L2]: “The moment problem is a classical question inanalysis, remarkable not only for its own elegance, but also for its extraordinaryrange of subjects theoretical and applied, which it has illuminated. From it flowdevelopments in function theory, in functional analysis, in spectral representationof operators, in probability and statistics, in Fourier analysis and the prediction ofstochastic processes, in approximation and numerical methods, in inverse problemsand the design of algorithms for simulating physical systems.” Looking at thedevelopments of the multidimensional moment problem over the last two decadesI would like to add real algebraic geometry, optimization, and convex analysis toLandau’s list.

This book is an advanced text on the moment problem on Rd and its moderntechniques. It is divided into four main parts, two devoted to one-dimensionalmoment problems and two others to multidimensional moment problems. In eachgroup we distinguish between full and truncated moment problems. Though ourmain emphasis is on real moment problems we include short treatments of themoment problem on the unit circle and of the complex moment problem.

Here is a brief description of the four parts.Part I deals with the one-dimensional full moment problem and develops

important methods and technical tools such as orthogonal polynomials, Jacobioperators, and Nevanlinna functions in great detail. Basic existence and uniquenesscriteria are obtained, but also a number of advanced results such as the Nevanlinnaparametrizations for indeterminate Hamburger and Stieltjes problems, finite ordersolutions, Nevanlinna–Pick interpolation, Krein and Friedrichs approximants ofStieltjes solutions, and others are included.

Part II is about one-dimensional truncated moment problems. The truncatedHamburger and Stieltjes moment problems are treated and Gauss’ quadratureformulas are derived. In the case of bounded intervals the classical theory ofMarkoff, Krein, and Akhiezer on canonical and principal measures, maximalmasses, and the moment cone are studied. Part II also contains a self-containeddigression to the trigonometric moment problem and some highlights of this theory(Schur algorithm, Verblunsky and Geronimus’ theorems).

In Part III the multidimensional full moment problem on closed semi-algebraicsubsets of Rd is investigated. Here real algebraic geometry and operator theoryon Hilbert space are the main tools. For compact semi-algebraic sets the interplaybetween strict Positivstellensätze and the moment problem leads to satisfactoryexistence results for the moment problem. In the case of closed semi-algebraic setsexistence criteria and determinacy are much more subtle. The fibre theorem andits applications and the multidimensional determinacy problem are investigated indetail. Further, we derive basic existence and determinacy results for the complex

6 Preface and Overview

moment problem. The two main Chaps. 12 and 13 are the heart of Part III andalso of the book. At the end of Part III we touch very briefly upon semidefiniteprogramming and applications of moment methods to polynomial optimization.

Part IV gives an introduction to the multidimensional truncated moment problem.Existence theorems in terms of positivity and the flat extension theorem are derived.Fundamental technical tools (Hankel matrices, evaluation polynomials, the apolarscalar product for homogeneous polynomials) are developed and discussed in detail.A number of important special topics (the core variety, maximal masses, deter-minacy, Carathéodory numbers) are also studied. The multidimensional truncatedmoment problem is an active topic of present research. It is expected that convexanalysis and algebraic geometry will provide new powerful methods and that thestatus of this area might essentially change in the coming decades. For this reason,we have not treated all recent developments; instead we have concentrated on basicresults and concepts and on selected special topics.

All moment problems treated in this book deal with integral representations oflinear functionals on a commutative unital algebra or on a (in most cases finite-dimensional) vector space of continuous functions on a locally compact space. Mostof them are moment problems on certain �-semigroups. In two introductionarychapters, general results on integral representations of positive functionals areobtained and notions concerning moment problems for �-semigroups are developed.These results and notions will play an essential role throughout the whole book.

As mentioned above, the main focus of this book is on the four versions of thescalar classical moment problem. In the course of this we develop fundamentalconcepts and technical tools and we derive deep classical theorems and very recentresults as well. Also, we present a number of new results and new proofs. In thisbook, we do not treat matrix moment problems, operator moment problems, infinite-dimensional moment problems, or noncommutative moment problems.

Apart from the two introductory chapters the parts of the book and a number ofchapters can be read (almost) independently from each other. Sometimes a technicalfact from another part is used in a proof; it can be filled easily. In order to beindependent from previous chapters we have occasionally repeated some notation.

Several courses and seminars on moment problems can be built on this book bychoosing appropriate material from various parts. Each course or seminar shouldprobably start with the corresponding results from Sects. 1.1 and 1.2. For a onesemester basic course on the one-dimensional moment problem this could befollowed by Sects. 2.1–2.2, Chap. 3, and the core material of the first sections ofChaps. 4–6. A one semester advanced course on the multidimensional momentproblem could be based on Sects. 11.1–11.3, 11.5–11.6, and selected material fromChaps. 12 and 13, avoiding technical subtleties. Here applications to optimizationfrom Chap. 15 are optional. Chapters 8, 9, 10, and 16 are almost self-contained andcould be used for special seminars on these topics. Most exercises at the end ofeach chapter should be solvable by active students; some of them contain additionalresults or information on the corresponding topics.

As the title of the book indicates, the real moment problem is the central topic.Our main emphasis is on a rigorous treatment of the moment problem, but also

Preface and Overview 7

on important methods and technical tools. Often several proofs or approaches tofundamental results are given. For instance, Hamburger’s theorem 3.8 is derivedin Sect. 3.2 from Haviland’s theorem, in Sect. 6.1 from the spectral theorem, and inSect. 9.2 from the truncated moment problem. This is also true for a number of otherresults such as Stieltjes’ theorem, Carleman’s theorem, Markov’s theorem, and thePositivstellensätze and moment problem theorems in Chap. 12. At various placesexplicit formulas in terms of the moments are provided even if they are not neededfor treating the moment problem. Large parts of the material, especially in Parts IIIand IV, appear for the first time in a book and a number of results are new.

Necessary prerequisities for this book are a good working knowledge of measureand integration theory and of polynomials, but also the basics of holomorphicfunctions in one variable, functional analysis, convex sets, and elementary topology.With this background about two thirds of the book should be readable by graduatestudents. In the remaining third (more precisely, in parts of Chaps. 4–7 and Part III)Hilbert space operator theory and real algebraic geometry play an essential role.The short disgression into real algebraic geometry given in Sect. 12.1 covers allthat is needed; for more details we refer to the standard books [Ms1] and [PD].Elementary facts on unbounded symmetric or self-adjoint operators and the spectraltheorem (multidimensional versions in Sect. 12.5 and 15.3) are used at variousplaces; necessary notions and facts are collected in Appendix A.7. In Chap. 8Friedrichs and Krein extensions of positive symmetric operators occur; they arebriefly explained in Sect. 8.1. All operator-theoretic notions and results needed inthis book can be found (for instance) in the author’s Graduate Text [Sm9].

In large parts of the book results and techniques from other mathematicalfields are used. In most cases we state or develop such results with referenceto the literature at the places where they are needed. Further, I have added sixappendices: on measure theory, on Pick functions and Stieltjes transforms, onpositive semidefinite matrices, on locally convex topologies, on convex sets, andon Hilbert space operators. These appendices collect notions and facts that are usedoften and at different places of the text. For some results we have included proofs.

Some general notation is collected after the table of contents. Though I tried toretain standard terminology in most cases, occasionally I have made some changes,we hope for the better. For instance, instead of the term “moment matrix” I preferred“Hankel matrix” and denoted it by Hn.L/ or H.L/: Also there is some overlappingnotation. While the symbol A is used for one of the four Nevanlinna functions inChap. 7 (following standard notation), it denotes a matrix or a Hilbert space operatorat other places. The meaning will be always clear from the context. The underlyingalgebras are usually denoted by sanserif letter such as A, B:

Continued fractions are avoided in this book (in Sect. 6.7 only the notion isbriefly explained). Instead I have put my emphasis on operator-theoretic approaches,because I am convinced that these methods are more promising and powerful, inparticular concerning the multidimensional case.

In writing this book I benefited very much from N.I. Akhiezer’s classic [Ak]and B. Simon’s article [Sim1], but also from standard books such as [BCRl], [KN],[KSt], [Ms1], [Sim3], [Chi1] and from the surveys [La2], [AK]. Applications and

8 Preface and Overview

ramifications of moment problems and related topics are discussed and developedin the AMS volume [L2] and in the books [Ls2], [BW].

I feel unable to give precise credits for all results occuring in this book. In theNotes at the end of each chapter I have given (to the best of my knowledge) creditsfor some results, including the main theorems, and some hints to the literature andfor further reading. In the bibliography I have listed some key classical papers.

I am indebted to B. Reznick, C. Scheiderer, and J. Stochel for valuable commentson parts of the manuscripts. I am grateful to Ph. di Dio for reading the whole textand for his helpful suggestions. Also, I should like to thank R. Lodh and A.-K.Birchley-Brun from Springer-Verlag for their help in publishing this book.

Leipzig, Germany Konrad SchmüdgenMay 7, 2017

General Notations

Numbers

N0 nonnegative integersN positive integersZ integersR real numbersC complex numbersT complex numbers of modulus oneD complex numbers of modulus less than oneRC D Œ0; C1/

CC D fz 2 C W Im z > 0gi complex unit

Spaces and Sets

Rd d-dimensional real spaceCd d-dimensional complex spacePd.R/ d-dimensional real projective spaceTd d-torusSd�1 unit sphere in Rd

Matrices

Mn;k.K/ .n; k/-matrices over K, Mn.K/ D Mn;n.K/

Symn symmetric matrices in Mn.R/

AT transposed matrix of A

Measures

MC.X / (positive) Radon measures on a locally compact Hausdorff space XM.X / complex Radon measures on XL1.X ; �/ �-integrable Borel functions on Xıx delta measure at the point x�M characteristic function of a set MRadon measures are always nonnegative!

9

10 General Notations

Polynomials

x˛ D x˛1

1 � � � x˛dd for ˛ D .˛1; � � � ; ˛d/ 2 Nd

0

RdŒx� D RŒx1; : : : ; xd�RdŒx�n D fp 2 RdŒx� W deg.p/ � ngHd;m homogeneous polynomials from RdŒx� of degree mqh homogenization of qPos.K/ D ff 2 E W f .x/ � 0; x 2 Kg, where E C.X IR/; K XPos.A;K/ D fp 2 A W p.x/ � 0; x 2 KgZ.p/ D fx 2 Rd W p.x/ D 0g for p 2 RdŒx�d is the number of variables and n;m denote the degrees of polynomials!

Moments, Moment Sequences, and Measures

s˛ D Rx˛ d� ˛-the moment of �

s D .s˛/ moment sequences.x/; sN.x/ moment vector of the delta measure ıxSmC1;S;S.A;K/ moment conesMC.Rd/ D f� 2 MC.Rd/ W R jx˛j d� < C1 for ˛ 2 Nd

0g Radon measureswith finite moments

Ms D f� 2 MC.Rd/ W s˛ D Rx˛ d� for ˛ 2 Nd

0g representing measures of sLs Riesz functional associated with s defined by Ls.x˛/ D s˛

Hn.s/ finite Hankel matrix associated with sH.s/ infinite Hankel matrix associated with sDn.s/ D detHn.s/ Hankel determinant

Orthogonal Polynomials and Functions

pn orthonormal polynomial of the first kindPn monic orthogonal polynomial of the first kindqn orthogonal polynomial of the second kindQn monic orthogonal polynomial of the second kindpz D .p0.z/; p1.z/; p2.z/; : : : /

qz D .q0.z/; q1.z/; q2.z/; : : : /

P Pick functions�s Friedrichs parameterA.z/;B.z/;C.z/;D.z/ entire functions in the Nevanlinna parametrization

Operators

X multiplication operator by the variable xJ infinite Jacobi matrixT D TJ Jacobi operator for the Jacobi matrix Jd D f.c0; c1; : : : ; cn; 0; 0; : : : / W cj 2 C; n 2 N0g finite complex sequenceshp; qis D Ls.pq/ scalar product on CŒx� associated with sHs Hilbert space completion of .CŒx�; h�; �is/X1; : : : ;Xd multiplication operators by the variables x1; : : : ; xd

General Notations 11

hp; qiL D L.pq/ scalar product on CdŒx� associated with LHL Hilbert space completion of .CdŒx�; h�; �iL/�L GNS representation associated with LTF Friedrichs extension of a positive symmetric operator TTK Krein extension of a positive symmetric operator T�.T/ spectrum of T.T/ resolvent set of TD.T/ domain of TN .T/ null space of TET spectral measure of T

Real Algebraic Geometry

T.f/ preordering generated by f, where f is a finite subset ofRdŒx�Q.f/ quadratic module generated by fK.f/ semi-algebraic set defined by fRŒV� algebra of regular functions on a real algebraic set VP2

n sum of squares p2 of polynomials p 2 RdŒx�, wheredeg p � n

OA characters of a real unital algebra APA2 sum of squares a2 of elements a of a real algebra A

AC D ff 2 A W �.f / � 0; � 2 OA gMoment Functionals and Related Sets

L;L.A;K/ cones of moment functionalslx point evaluation functional at xLg D L.g�/ localization of L at gML D f� 2 MC.Rd/ W L.p/D R

p.x/ d� for p2RdŒx�g representing measures of LL.t/ maximal mass of representing measures of L at tH.L/ Hankel matrix of LW.L/;W.L/ set of atoms of representing measures of LV.L/;V.L/ core variety of LNL D ff 2 A W L.fg/ D 0; g 2 A gVL D ft 2 Rd W f .t/ D 0; f 2 NL gNC.L;K/ D ff 2 Pos.A;K/ W L.f / D 0 gVC.L;K/ D ft 2 Rd W f .t/ D 0; f 2 NC.L;K/ gNC.L/ D NC.L;Rd/

VC.L/ D V.L;Rd/

NC.L/ D ff 2 EC W L.p/ D 0 gVC.L/ D fx 2 X W f .x/ D 0; f 2 NC.L/ gSets are denoted by braces such as fxi W i 2 Ig, while sequences are written as.xn/n2N or .xn/.