# Flat Extensions of Positive Moment Matrices: Recursively Generated Relations

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*Raúl E. Curto; Lawrence A. Fialkow*

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.

#### Table of Contents

# Table of Contents

## Flat Extensions of Positive Moment Matrices: Recursively Generated Relations

- Contents vii8 free
- Chapter 1. Introduction 112 free
- Chapter 2. Flat Extensions for Moment Matrices 1122 free
- Chapter 3. The Singular Quartic Moment Problem 2536
- Chapter 4. The Algebraic Variety of γ 3243
- Chapter 5. J.E. McCarthy's Phenomenon and the Proof of Theorem 1.5 4455
- Summary of Results 4960
- Bibliography 5263
- List of Symbols 5566