Moment vanishing problem and positivity: some examples
Let \(\Omega\) be a compact domain in \(\mathbb R^n\), let \(\mu\) be an arbitrary measure on \(\Omega\), and let \(P,\,q\) be complex continuous functions on \(\Omega\). The moment vanishing problem means to give necessary and sufficient conditions for vanishing of all moments of the form \(m_k(P,q,\mu)=\int_\Omega P^k(x)q(x)d\mu(x),\,k\geq1\) integer. When \(\Omega\) is a semi-algebraic compact set and \(P,\,q\) are polynomials, the moment vanishing problem is called the \textit{polynomial moment vanishing problem}. One of the authors (W. Zhao) previously stated the following conjecture: For positive \(\mu\) and \(P,q\) polynomials, if \(m_k(P,1,\mu)=0\) for all \(k\geq1\), then \(m_k(P,q,\mu)=0\) for \(k\gg1\) for any \(q\). The authors provide, in some special cases, necessary and sufficient conditions on the measure \(\mu\) for which the conjecture stated above holds.
scientific article; zbMATH DE number 5860018
scientific article