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Closure of the Cone of Sums of $2d$-powers in Certain Weighted $\ell_1$-seminorm Topologies

Published:2012-12-29

• Mehdi Ghasemi,
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N 5E6
• Murray Marshall,
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N 5E6
• Sven Wagner,
Technische Universität Dortmund, Fakultät für Mathematik, Lehrstuhl VI, Vogelpothsweg 87, 44227 Dortmund, Germany
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Abstract

In a paper from 1976, Berg, Christensen and Ressel prove that the closure of the cone of sums of squares $\sum \mathbb{R}[\underline{X}]^2$ in the polynomial ring $\mathbb{R}[\underline{X}] := \mathbb{R}[X_1,\dots,X_n]$ in the topology induced by the $\ell_1$-norm is equal to $\operatorname{Pos}([-1,1]^n)$, the cone consisting of all polynomials which are non-negative on the hypercube $[-1,1]^n$. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $\ell_1$-seminorm topology associated to an absolute value. In the present paper we give a new proof of these results which is based on Jacobi's representation theorem from 2001. At the same time, we use Jacobi's representation theorem to extend these results from sums of squares to sums of $2d$-powers, proving, in particular, that for any integer $d\ge 1$, the closure of the cone of sums of $2d$-powers $\sum \mathbb{R}[\underline{X}]^{2d}$ in $\mathbb{R}[\underline{X}]$ in the topology induced by the $\ell_1$-norm is equal to $\operatorname{Pos}([-1,1]^n)$.
 Keywords: positive definite, moments, sums of squares, involutive semigroups
 MSC Classifications: 43A35 - Positive definite functions on groups, semigroups, etc. 44A60 - Moment problems 13J25 - Ordered rings [See also 06F25]