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The truncated moment problem on curves $y = q(x)$ and $yx^\ell = 1$Aljaž Zalar, 2024, original scientific article
Abstract: In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form $y = q(x), q(x) \in \mathbb{R} [x], \deg q ≥ 3$ and $yx^\ell = 1, \ell \in \mathbb{N}$ \ $\{1\}$. For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety $y = x^3$. Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety $y = x^3$. Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: 10.1007/s00020-021-02628-6.; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. doi.org/10.1016/j.laa.2022.05.008.; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: 10.1016/j.jmaa.2022. 126563.], and improve Fialkow’s bound to $\deg q − 1$ (resp. $\ell + 1$) for curves $y = q(x)$ (resp. $yx^\ell = 1$). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are $k \deg q$ (resp. $k(\ell+ 1)$) for curves $y = q(x)$ (resp. $yx^\ell = 1$) for even degree sequences, while for odd ones they are $k \deg q − \bigl \lceil \frac{\deg q}{2} \bigr \rceil$ (resp. $k(\ell + 1) − \bigl \lfloor \frac{\ell}{2} \bigr \rfloor + 1$) for curves $y = q(x)$ (resp. $yx^\ell = 1$). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves $y = x^\ell, \ell = 2, 3,$ and add a new solvability condition to the even degree case on the curve $y = x^2$.
Keywords: truncated moment problems, K-moment problems, K-representing measure, minimal measure, moment matrix extensions, positivstellensatz, linear matrix inequality
Published in DiRROS: 25.07.2024; Views: 104; Downloads: 98
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