Non-extremal sextic moment problems

Curto, Raúl E.; Yoo, Seonguk

doi:10.1016/j.jfa.2015.04.014

Cited by 14 publications

(8 citation statements)

References 39 publications

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“…Therefore, only the cases m = 1, 2, 3 and 4 (the quadratic, the cubic and the quartic moment problem) have been completely achieved. All the other cases (quintic, sixtic, ...) are open and interest several authors; as indicated in many recent papers (see, for instance, [13,15,16,37,38]).…”

Section: Introductionmentioning

confidence: 97%

The quintic complex moment problem

Azhar¹,

Harrat²,

Idrissi³

et al. 2019

Operators and Matrices

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Let γ (m) ≡ {γ ij } 0≤i+j≤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 µ on C (called a representing measure for γ (m) ) such that γ ij = z i z j dµ for 0 ≤ i + j ≤ 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.2010 Mathematics Subject Classification. Primary 47A57, 15A83; Secondary 30E05, 44A60.

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Section: Introductionmentioning

confidence: 97%

The quintic complex moment problem

Azhar¹,

Harrat²,

Idrissi³

et al. 2019

Operators and Matrices

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“…So it remains to study the case y 2 = 1 or equivalently the TMP on the union of two parallel lines (TMP-2pl). Concerning the TMP on the varieties beyond the quadratic ones the solutions typically require testing some additional numerical conditions which depend on given moments (see [CY14,CY15,Yoo17a,Yoo17b]).…”

Section: Introductionmentioning

confidence: 99%

The truncated moment problem on the union of parallel lines

Zalar

2021

Preprint

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In this article we study the bivariate truncated moment problem (TMP) of degree 2k for the union of parallel lines. First we present an alternative proof of Fialkow's solution [Fia15] of the TMP for the union of two parallel lines (TMP-2pl) using the solution of the truncated Hamburger moment problem (THMP). We add a new equivalent solvability condition, which is then used together with the THMP, to solve the TMP for the union of three parallel lines (TMP-3pl), which is our second main result. Finally, we establish a sufficient condition for the existence of a measure for the TMP on the union of n parallel lines in the pure case, in which the moment matrix M k is of the highest possible rank, i.e., the only column relations come from the union of n lines. The condition is based on the feasibility of a certain linear matrix inequality, corresponding to the extension of M k by adding rows and columns indexed by some monomials of degree k + 1. The proof is by induction on n, where n ≥ 4 and for the base of induction n = 4 we use the solutions of the TMP-3pl and the THMP.

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“…Using the flat extension theorem as the main tool the 2-dimensional TMP has been concretely solved in the following cases: K is the variety defined by a polynomial p(x, y) = 0 Fia11], M(k) has a special feature called recursive determinateness [CF13] and in the extremal case with the equality in the variety condition [CFM08]. Some other special cases have been solved in [Kim14, CS15,Fia17,Ble15,BF20]. In [Fia11], Fialkow studied also the TMP for the curves of the form y = g(x) and yg(x) = 1, where g ∈ R[x] is a polynomial, and obtained the bound on the degree m for which the existence of a positive extension M(m) of M(k) is equivalent to the existence of a measure.…”

Section: Introductionmentioning

confidence: 99%

The strong truncated Hamburger moment problem with and without gaps

Zalar¹

2021

Preprint

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The strong truncated Hamburger moment problem (STHMP) of degree (−2k 1 , 2k 2 ) asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on R \ {0}, such thatUsing the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we solve the STHMP. Then, using the equivalence with the STHMP of degree (−2k, 2k), we obtain the solution of the 2-dimensional truncated moment problem (TMP) of degree 2k with variety xy = 1, first solved by Curto and Fialkow [CF05]. Our addition to their result is the fact previously known only for k = 2, that the existence of a measure is equivalent to the existence of a flat extension of the moment matrix. Further on, we solve the STHMP of degree (−2k 1 , 2k 2 ) with one missing moment in the sequence, i.e., β −2k1+1 or β 2k2−1 , which also gives the solution of the TMP with variety x 2 y = 1 as a special case, first studied by Fialkow in [Fia11].

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Non-extremal sextic moment problems

Cited by 14 publications

References 39 publications

The quintic complex moment problem

The quintic complex moment problem

The truncated moment problem on the union of parallel lines

The strong truncated Hamburger moment problem with and without gaps

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