The Cubic Complex Moment Problem

Let ({s = {s_{jk}}_{0 leq j+k leq 3}}) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure ({sigma}) on ({mathbb{C}}) (called a representing measure for s) such that ({s_{jk} = int_{mathbb{C}}bar{z}^j z^k dsigma(z)}) for ({0 leq j + k leq 3}) . Put $$Phi = left(begin{array}{lll} s_{00} & s_{01} & s_{10} s_{10} & s_{11} & s_{20} s_{01} & s_{02} & s_{11}end{array}right), quad Phi_z = left(begin{array}{lll}s_{01} & s_{02} & s_{11} s_{10} & s_{12} & s_{21} s_{02} & s_{03} & s_{12}end{array} right)quad {rm and}quadPhi_{bar{z}} = (Phi_z)^*.$$ If ({Phi succ 0}) , then the commutativity of ({Phi^{-1} Phi_z}) and ({Phi^{-1} Phi_{bar{z}}}) is necessary and sufficient for the existence a 3-atomic representing measure for s. If ({Phi^{-1} Phi_z}) and ({Phi^{-1} Phi_{bar{z}}}) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set ({K subseteq mathbb{C}}) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure ({sigma}) which satisfies ({{rm supp} sigma cap K neq emptyset}) or ({{rm supp} sigma subseteq K}) . The cases when ({K = overline{mathbb{D}}}) and ({K = mathbb{T}}) are considered in detail.