eBook ISBN:  9781470402372 
Product Code:  MEMO/136/648.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $27.00 
eBook ISBN:  9781470402372 
Product Code:  MEMO/136/648.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $27.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 136; 1998; 56 ppMSC: Primary 47; 44; 30; Secondary 15;
In this book, the authors develop new computational tests for existence and uniqueness of representing measures \(\mu\) in the Truncated Complex Moment Problem: \(\gamma _{ij}=\int \bar z^iz^j\, d\mu\) \((0\le i+j\le 2n)\).
Conditions for the existence of finitely atomic representing measures are expressed in terms of positivity and extension properties of the moment matrix \(M(n)(\gamma )\) associated with \(\gamma \equiv \gamma ^{(2n)}\): \(\gamma_{00}, \dots ,\gamma _{0,2n},\dots ,\gamma _{2n,0}\), \(\gamma _{00}>0\). This study includes new conditions for flat (i.e., rankpreserving) extensions \(M(n+1) \) of \(M(n)\ge 0\); each such extension corresponds to a distinct rank \(M(n)\)atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consistency, and normal consistency, the existence problem for minimal representing measures is reduced to the solubility of small systems of multivariable algebraic equations. In a variety of applications, including cases of the quartic moment problem (\(n=2\)), the text includes explicit contructions of minimal representing measures via the theory of flat extensions. Additional computational texts are used to prove nonexistence of representing measures or the nonexistence of minimal representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higherdimensional moment problems that do not appear in the classical onedimensional moment problem.
ReadershipGraduate students and research mathematicians working in operator theory.

Table of Contents

Chapters

1. Introduction

2. Flat extensions for moment matrices

3. The singular quartic moment problem

4. The algebraic variety of $\gamma $

5. J.E. McCarthy’s phenomenon and the proof of Theorem 1.5


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In this book, the authors develop new computational tests for existence and uniqueness of representing measures \(\mu\) in the Truncated Complex Moment Problem: \(\gamma _{ij}=\int \bar z^iz^j\, d\mu\) \((0\le i+j\le 2n)\).
Conditions for the existence of finitely atomic representing measures are expressed in terms of positivity and extension properties of the moment matrix \(M(n)(\gamma )\) associated with \(\gamma \equiv \gamma ^{(2n)}\): \(\gamma_{00}, \dots ,\gamma _{0,2n},\dots ,\gamma _{2n,0}\), \(\gamma _{00}>0\). This study includes new conditions for flat (i.e., rankpreserving) extensions \(M(n+1) \) of \(M(n)\ge 0\); each such extension corresponds to a distinct rank \(M(n)\)atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consistency, and normal consistency, the existence problem for minimal representing measures is reduced to the solubility of small systems of multivariable algebraic equations. In a variety of applications, including cases of the quartic moment problem (\(n=2\)), the text includes explicit contructions of minimal representing measures via the theory of flat extensions. Additional computational texts are used to prove nonexistence of representing measures or the nonexistence of minimal representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higherdimensional moment problems that do not appear in the classical onedimensional moment problem.
Graduate students and research mathematicians working in operator theory.

Chapters

1. Introduction

2. Flat extensions for moment matrices

3. The singular quartic moment problem

4. The algebraic variety of $\gamma $

5. J.E. McCarthy’s phenomenon and the proof of Theorem 1.5