The Cubic Complex Moment Problem

Let (Formula presented.) 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 (Formula presented.) on (Formula presented.) (called a representing measure for s) such that (Formula presented.) for (Formula presented.), then the commutativity of (Formula presented.)is necessary and sufficient for the existence a 3-atomic representing measure for s.(Formula presented.)If Φ Φ 0, then the commutativity of Φ −1Φz and Φ −1Φ¯z is necessary and sufficient for the existence a 3-atomic representing measure for s. If Φ −1Φz and Φ −1Φ¯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 ࣮ C necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure σwhich satisfies supp σ ∩ K ≠= ∅ or supp σ ࣮ K. The cases when K = D and K = T are considered in detail..