Flat extensions of positive moment matrices: recursively generated relations
Raúl E. Curto, Lawrence A. Fialkow · Memoirs of the American Mathematical Society · 1998
We develop new computational tests for existence and uniqueness of representing measures µ in the Truncated Complex Moment Problem:We characterize the existence of finitely atomic representing measures in terms of positivity and extension properties of the moment matrix M (n)(γ) associated with γ ≡ γ (2n) : γ 00 , . . ., γ 0,2n , . . ., γ 2n,0 , γ 00 > 0 (Theorem 1.5).We study conditions for flat (i.e., rank-preserving) extensions M (n + 1) of M (n) ≥ 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, we reduce the existence problem for minimal representing measures to the solubility of small systems of multivariable 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 representing measures.In other examples, we use these tests to demonstrate the non-existence of representing measures or the non-existence of minimal representing measures.Key words and phrases.Truncated complex moment problem, moment matrix extension block, flat extensions of positive matrices, recursively generated relations, algebraic variety of a moment sequence. ContentsChapter 1. Introduction Chapter 2. Flat Extensions for Moment Matrices Chapter 3. The Singular Quartic Moment Problem Chapter 4. The Algebraic Variety of γ Chapter 5. J.E. McCarthy's Phenomenon and the Proof of Theorem 1.