The quintic complex moment problem

Abstract

Let ${\gamma }^{(m)}\equiv \{{\gamma }_{ij}{\}}_{0\le i+j\le m}$ be a given complex-valued sequence. The truncated complex moment problem (TCMP in short) involves determining necessary and sufficient conditions for the existence of a positive Borel measure $\mu$ on $\mathbb{C}$ (called a representing measure for ${\gamma }^{(m)}$) such that ${\gamma }_{ij}=\int {\overline{z}}^i{z}^{j}d\mu$ for $0\le i+j\le m$. The TCMP has been completely solved only when $m=1,2,3,4$. We provide in this paper a concrete solution to the quintic TCMP (that is, when $m=5$). We also study the cardinality of the minimal representing measure. Based on the bivariate recurrences sequences's properties with some Curto-Fialkow's results, our method intended to be useful for all odd-degree moment problems.

16 pages

References in corpus (4)

• Truncated K-moment problems in several variables
• Concrete Solution to the Nonsingular Quartic Binary Moment Problem
• Non-extremal sextic moment problems
• The Division Algorithm in Sextic Truncated Moment Problems