The Cubic Complex Moment Problem

Abstract

Let s = {s jk } 0≤j+k≤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 σ on C (called a representing measure for s) such that s jk = CzIf Φ 0, then the commutativity of Φ −1 Φz and Φ −1 Φz is necessary and sufficient for the existence a 3-atomic representing measure for s. If Φ −1 Φz and Φ −1 Φz 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 σ which satisfies supp σ ∩ K = ∅ or supp σ ⊆ K. The cases when K = D and K = T are considered in detail.Mathematics Subject Classification. Primary 47A57; Secondary 30E05, 44A60.

A new approach to the nonsingular cubic binary moment problem

“…As we have indicated before, the nontrivial cases of the cubic binary moment problem arise when the submatrix M(1) of β (3) is nonsingular. Moreover, as noted in [12] the positive semidefiniteness of M( 1) is always a necessary condition for the existence of a representing measure. Thus, in the sequel we focus on cubic binary moment problems with M(1) positive definite.…”

“…a 0 a 1 + a 1 a 2 1 + a 0 a 2 + a 1 a 3 a 0 a 1 + a 1 a 2 a 2 1 + a 2 2 a 1 a 2 + a 2 a 3 1 + a 0 a 2 + a 1 a 3 a 1 a 2 + a 2 a 3 1 + a 2 2 + a 2 3   . Condition (3.7) is equivalent to the commutativity of the matrices defined in [12].…”

On A Class of Real Quintic Moment Problem

“…There is an equivalent to the TRMP, that is the TCMP (truncated complex moment problem) [6], hence we use the term (TMP) problem of shortened moments. The multidimensional truncated moment problem has been the subject of several studies, mainly by Curto, Fialkow and others as found for example in [3,4,5,6,7,8,10,11,12,13,16,18,23,24,25]. In 1994 J. Stochel [23] showed that the truncated moment problem is more general than the full moment problem, i.e.…”

“…For n = 3, it has been closely investigated and in particular the extreme case where the rank of the M (3) matrix of moments associated to β (6) and the cardinal of the associated core variety are equal [8,10,12,24,25]. For m = 3, we can find a complete solution in [18] based on the commutativity conditions of the matrix associated to the sequence of cubic moment. In [13], R. Curto and S. Yoo presented an alternative solution for the non-singular cubic moment problem.…”

The truncated moment problem on the union of parallel lines

“…Among other cases of the TMP let us mention the recent core variety approach which yielded important new results on the TMP [Fia17,DS18,BF20]. For the solution of the cubic TMP see [Kim14]. For some other results and variants of the TMP see also [Vas03,BK12,Nie14,Ble15,IKLS17,Kim21,CGIK+].…”