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.