Application of localization to the multivariate moment problem II


  • Murray Marshall



The paper is a sequel to the paper [5], Math. Scand. 115 (2014), 269--286, by the same author. A new criterion is presented for a PSD linear map $L \colon \mathbb{R}[\underline{x}] \to \mathbb{R}$ to correspond to a positive Borel measure on $\mathbb{R}^n$. The criterion is stronger than Nussbaum's criterion (Ark. Math. 6 (1965), 171--191) and is similar in nature to Schmüdgen's criterion in Marshall [5] and Schmüdgen, Ark. Math. 29 (1991), 277--284. It is also explained how the criterion allows one to understand the support of the associated measure in terms of the non-negativity of $L$ on a quadratic module of $\mathbb{R}[\underline{x}]$. This latter result extends a result of Lasserre, Trans. Amer. Math. Soc. 365 (2013), 2489--2504. The techniques employed are the same localization techniques employed already in Marshall (Cand. Math. Bull. 46 (2003), 400--418, and [5]), specifically one works in the localization of $\mathbb{R}[\underline{x}]$ at $p = \prod_{i=1}^n(1+x_i^2)$ or $p' = \prod_{i=1}^{n-1}(1+x_i^2)$.


Berg, C. and Christensen, J. P. R., Exposants critiques dans le problème des moments, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 15, 661–663.

Fuglede, B., The multidimensional moment problem, Exposition. Math. 1 (1983), no. 1, 47–65.

Lasserre, J. B., The $mathrm K$-moment problem for continuous linear functionals, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2489–2504.

Marshall, M., Approximating positive polynomials using sums of squares, Canad. Math. Bull. 46 (2003), no. 3, 400–418.

Marshall, M., Application of localization to the multivariate moment problem, Math. Scand. 115 (2014), no. 2, 269–286.

Nussbaum, A. E., Quasi-analytic vectors, Ark. Mat. 6 (1965), 179–191.

Schmüdgen, K., On determinacy notion for the two-dimensional moment problem, Ark. Mat. 29 (1991), no. 2, 277–284.

Sodin, M., A note on the Hall-Mergelyan theme, Mat. Fiz. Anal. Geom. 3 (1996), no. 1-2, 164–168.



How to Cite

Marshall, M. (2017). Application of localization to the multivariate moment problem II. MATHEMATICA SCANDINAVICA, 120(1), 124–128.