Resumen de The complex moment problem: determinacy and extendibility

Dariusz Cichon, Jan Stochel, Franciszek Hugon Szafraniec

• 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 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.