The truncated Hamburger moment problems with gaps in the index set (Q2032313)

The truncated Hamburger moment problem examined in this paper considers moments \(\beta_0,\ldots,\beta_{2k}\) of a measure supported on a curve \(K\) in \(\mathbb{R}^2\) described by a simple polynomial \(p(x,y)=0\). Such problems were considered before in several papers by Curto, Fialkow, and their coworkers; for example, in [\textit{L. A. Fialkow}, Trans. Am. Math. Soc. 363, No. 6, 3133--3165 (2011; Zbl 1220.47021)], the polynomial is \(p(x,y)=y-x^3\). Then only three columns in the table of the bivariate moments are needed because the remaining ones can be recursively generated. In that way, the moment sequence can be analysed using the classical setting of \(K=\mathbb{R}\) using scalar moments. Fialkow showed that when these generate positive (semi) definite Hankel matrices with certain rank conditions, then an (atomic) representing measure exists. In the present paper, the extra complication is that one or two of the moments are unknown (they are the gaps in the sequence). The following cases are considered: (1) \(p(x,y)=y-x^3\) with missing \(\beta_{2k-1}\); (2) \(p(x,y)=y-x^4\) with missing \(\beta_{2k-2},\beta_{2k-1}\) but \(\beta_{3,2k-2}=\int_K x^3y^{2k-2}d\mu\) is known (3) \(p(x,y)=y^2-x^3\) with missing \(\beta_1\); (4) \(p(x,y)=y^3-x^4\) with missing \(\beta_1,\beta_2\) but \(\beta_{5/3,0}=\int_K x^{5/3}y^0 d\mu\) is known. In (2) and (4), two bivariate moments are missing, but then an extra partial moment is needed to keep the columns recursively generated. The approach is based on Hankel matrices with unknown entries that satisfy already some partial conditions of being positive (semi) definite. It is shown when values for the unknown entries exist that generate positive (semi) definite Hankels satisfying the Fialkow conditions. Since the missing entries are near the beginning or near the end of the sequence, this can be obtained making use of \(2\times2\) or \(3\times3\) Schur complements in the Hankel matrix.