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Article overview
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Closure of the cone of sums of 2d-powers in certain weighted $ell_1$-seminorm topologies | Mehdi Ghasemi
; Murray Marshall
; Sven Wagner
; | Date: |
1 Sep 2011 | Abstract: | Berg, Christensen and Ressel prove that the closure of the cone of sums of
squares in the ring of real polynomials in the topology induced by the
$ell_1$-norm is equal to 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 which is valid for any commutative semigroup. In later work
Berg and Maserick and also 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. 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 $d>0$, the closure of the cone of
sums of 2d-powers in the ring of real polynomials in the topology induced by
the $ell_1$-norm is equal the cone consisting of all polynomials which are
non-negative on the hypercube $[-1,1]^n$. | Source: | arXiv, 1109.0048 | Services: | Forum | Review | PDF | Favorites |
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