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Cubic column relations in truncated moment problems
Journal article   Open access   Peer reviewed

Cubic column relations in truncated moment problems

Raúl E Curto and Seonguk Yoo
Journal of functional analysis, Vol.266(3), pp.1611-1626
02/01/2014
DOI: 10.1016/j.jfa.2013.11.024
url
https://doi.org/10.1016/j.jfa.2013.11.024View
Published (Version of record) Open Access

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 rankM(n)⩽cardVγ 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 (rankM(n)=cardVγ), 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 rankM(3)=7; we then apply the above mentioned result to obtain a concrete, computable, necessary and sufficient condition for the existence of a representing measure.
 Algebraic variety Riesz functional Harmonic polynomial Truncated moment problem Cubic column relation

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