The complex moment problem: determinacy and extendibility


  • Dariusz Cichoń
  • 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 $\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.


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How to Cite

Cichoń, D., Stochel, J., & Szafraniec, F. H. (2019). The complex moment problem: determinacy and extendibility. MATHEMATICA SCANDINAVICA, 124(2), 263–288.