TY - JOUR
AU - Dariusz Cichoń
AU - Jan Stochel
AU - Franciszek Szafraniec
PY - 2019/06/17
Y2 - 2020/06/01
TI - The complex moment problem: determinacy and extendibility
JF - MATHEMATICA SCANDINAVICA
JA - MathScand
VL - 124
IS - 2
SE - Articles
DO - 10.7146/math.scand.a-112091
UR - https://www.mscand.dk/article/view/112091
AB - Complex moment sequences are exactly those which admit positive definite extensions on the integer lattice points of the upper diagonal half-plane. Here we prove that the aforesaid extension is unique provided the complex moment sequence is determinate and its only representing measure has no atom at $0$. The question of converting the relation is posed as an open problem. A partial solution to this problem is established when at least one of representing measures is supported in a plane algebraic curve whose intersection with every straight line passing through $0$ is at most one point set. Further study concerns representing measures whose supports are Zariski dense in $\mathbb{C} $ as well as complex moment sequences which are constant on a family of parallel “Diophantine lines”. All this is supported by a bunch of illustrative examples.
ER -