FLAT EXTENSIONS OF POSITIVE MOMENT MATRICES - RECURSIVELY GENERATED RELATIONS

We develop new computational tests for existence and uniqueness of rep resenting measures mu in the Truncated Complex Moment Problem: (TCMP) gamma(ij) = integral (z) over bar(i)z(j) d mu (0 less than or equal to i + j less than or equal to 2n). We characterize the existence of fin itely atomic representing measures in terms of positivity and extensio n properties of the moment matrix M(n)(gamma) associated with gamma = gamma((2n)): gamma(00),..., gamma(0,2n),..., gamma(2n,0), gamma(00) > 0 (Theorem 1.5). We study conditions for flat (i.e., rank-preserving) extensions M(n + 1) of M(n) greater than or equal to 0; each such exte nsion 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 matr ices satisfying the tests of recursive generation, recursive consisten cy, and normal consistency, we reduce the existence problem for minima l representing measures to the solubility of small systems of multivar iable algebraic equations (Theorem 2.7). In a variety of applications, including cases of the quartic moment problem (n = 2; Theorem 1.10), we apply these tests so as to construct flat extensions and minimal re presenting measures. In other examples, we use these tests to demonstr ate the non-existence of representing measures or the non-existence of minimal representing measures.