Cubic column relations in truncated moment problems-英文资料.pdf

Available online at ScienceDirect Journal of Functional Analysis 266 (2014) 1611–1626 /locate/jfa Cubic column relations in truncated moment problems Raúl E. Curto ∗,1, Seonguk Yoo 2 Department of Mathematics, The University of Iowa, Iowa City, IA 52242, United States Received 13 April 2013; accepted 28 November 2013 Available online 19 December 2013 Communicated by S. Vaes Abstract For the truncated moment problem associated to a complex sequence γ (2n) = {γij }i,j ∈Z+ , i+j 2n to have a representing measure μ, it is necessary for the moment matrix M(n) to be positive semidefinite, and for the algebraic variety Vγ to satisfy rank M(n) card Vγ as well as a consistency condition: the Riesz functional vanishes on every polynomial of degree at most 2n that vanishes on Vγ . In previous work with L. Fialkow and H.M. Möller, the first named author proved that for the extremal case (rank M(n) = card Vγ ), positivity and consistency are sufficient for the existence of a representing measure. In this paper we solve the truncated moment problem for cubic column relations in M(3) of the form Z3 ¯ = itZ + uZ (u, t ∈ R); we do this by checking consistency. For (u, t) in the open cone determined by 0 |u| t 2 |u|, we first prove that the algebraic variety has exactly 7 points and rank M(3) = 7; we then apply the above mentioned result to obtain a concrete, computable, necessary and sufficient condition for the existence of a representing measu