Flat extensions of positive moment matrices : recursively generated relations
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Bibliographic Information
Flat extensions of positive moment matrices : recursively generated relations
（Memoirs of the American Mathematical Society, no. 648）
American Mathematical Society, 1998
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"November 1998, volume 136, number 648 (third of 6 numbers)." -- t.p.
Includes bibliographical references
Description and Table of Contents
Description
In this book, the authors develop new computational tests for existence and uniqueness of representing measures $\mu$ in the Truncated Complex Moment Problem: $\gamma_{ij}=\int \bar z^iz^j\, d\mu$ $(0\1e i+j\1e 2n)$. 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.
Table of Contents
Introduction Flat extensions for moment matrices The singular quartic moment problem The algebraic variety of $\gamma$ J. E. McCarthy's phenomenon and the proof of Theorem 1.5 Summary of results Bibliography List of symbols.
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