Real Algebraic and Analytic Geometry |

Closure of the cone of sums of 2d powers in certain weighted \ell_1-seminorm topologies.

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Submission: 2011, September 1.

*Abstract:
In 1976 Berg, Christensen and Ressel prove that the closure of the cone of sums of squares in the polynomial ring R[X_1,\dots,X_n] in the topology induced by the \ell_1-norm is equal to 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, also 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 proved in 2000. 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 in R[X_1,\dots,X_n] in the topology induced by the \ell_1-norm is equal to Pos([-1,1]^n). .*

Mathematics Subject Classification (2000): 43A35 Secondary 44A60.

Keywords and Phrases: positive definite, moments, sums of squares, involutive semigroups.

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