The \(K\)-moment problem for continuous linear functionals (Q2838073)

The author considers semi-algebraic sets \(K\subset\mathbb{R}^n\), that is, subsets of the form \(K=\{{\mathbf x}\in\mathbb{R}^n:p_j({\mathbf x})\geq0,~ j=0,1,\ldots,m\}\), where \((g_j)_{j=0}^m\) is a (finite) family of real polynomials on \(\mathbb{R}^n\) with \(g_0=1\). Such a set is clearly closed but not necessarily compact. The aim of the paper is to solve the \(K\)-moment problem for certain continuous linear functionals. More precisely, the algebra of all polynomials \(\mathbb{R}[{\mathbf x}]\) is endowed with a norm induced by the sequence \({\mathbf w}= (w_\alpha)_{\alpha\in\mathbb{N}^n}\), where \(w_\alpha=2\lceil|\alpha|/2\rceil!\). If \(f=\sum_{\alpha\in\mathbb{N}^n}f_\alpha{\mathbf x}^\alpha\) is an arbitrary polynomial, what the author calls the \(\ell_{\mathbf w}\)-norm of \(f\) is the quantity given by \(\sum_\alpha w_\alpha| f_\alpha|\).NEWLINENEWLINEFor a semi-algebraic set \(K\subset\mathbb{R}^n\) defined by the family of polynomials \(\{p_0,p_1,\break\ldots,p_m\}\), with \(p_0=1\), and for a linear functional \(L\) on \(\mathbb{R}[{\mathbf x}]\) which is \(\ell_{\mathbf w}\)-continuous, there exists a finite positive Borel measure \(\mu\) on \(K\) such that \(L(f)=\int_Kfd\mu\) for all polynomials \(f\in\mathbb{R}[{\mathbf x}]\) if and only if \(L(h^2g_j)\geq 0\) for all \(h\in \mathbb{R}[{\mathbf x}]\), \(j=0,1,\ldots,m\), and \(\sup_{\alpha\in\mathbb{N}^n}| L({\mathbf x}^\alpha)|/w_\alpha<\infty\). The main ingredient of the proof is the use of Carleman's condition, insured by the choice of the sequence \(\mathbf w\). Other related results are also obtained, exploiting the \(\ell_{\mathbf w}\)-continuity. In particular, with the notation from above, a polynomial \(f\) is positive on the support of \(\mu\) if and only if \(\int h^2fd\mu\geq0\) for all \(h\in \mathbb{R}[{\mathbf x}]\).