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Flat Extensions of Positive Moment Matrices: Recursively Generated Relations
 
Raúl E. Curto University of Iowa, Iowa City, IA
Lawrence A. Fialkow State University of New York, New Paltz, NY
Flat Extensions of Positive Moment Matrices: Recursively Generated Relations
eBook ISBN:  978-1-4704-0237-2
Product Code:  MEMO/136/648.E
List Price: $45.00
MAA Member Price: $40.50
AMS Member Price: $27.00
Flat Extensions of Positive Moment Matrices: Recursively Generated Relations
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Flat Extensions of Positive Moment Matrices: Recursively Generated Relations
Raúl E. Curto University of Iowa, Iowa City, IA
Lawrence A. Fialkow State University of New York, New Paltz, NY
eBook ISBN:  978-1-4704-0237-2
Product Code:  MEMO/136/648.E
List Price: $45.00
MAA Member Price: $40.50
AMS Member Price: $27.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1361998; 56 pp
    MSC: 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., rank-preserving) 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 non-existence of representing measures or the non-existence of minimal representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higher-dimensional moment problems that do not appear in the classical one-dimensional moment problem.

    Readership

    Graduate 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1361998; 56 pp
MSC: 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., rank-preserving) 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 non-existence of representing measures or the non-existence of minimal representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higher-dimensional moment problems that do not appear in the classical one-dimensional moment problem.

Readership

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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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