Extremal values of moments of nonnegative polynomials (Q2210421)

Let \(w\) be a nontrivial, nonnegative, and integrable weight function on \([0,1]\) and consider \[ m_k(p):=\int_{0}^{1}x^kp(x)w(x)dx, \] the \(k\)th moment of a polynomial \(p\) on \([0,1]\). For \(n=2m-2\), let \(P_n^{(\lambda)}(x)\) be the orthonormal polynomial of degree \(n\) on \([0,1]\) with respect to the weight \[ w(x,\lambda):=w(x)\sum_{k=0}^{m-1}\lambda^{m-1-k}x^k\geq0 \] depending on a parameter \(\lambda\in[0,1]\). Moreover, let \(\bigl\{t_{1,n}(\lambda),\ldots,t_{n,n}(\lambda)\bigr\}\) be the zeros of \(P_n^{(\lambda)}(x)\) numbered in decreasing order. The first main result of the paper is that there exist positive numbers \(\{\delta_i^1\}_{i=2}^{m}\) and \(\{\delta_i^2\}_{i=1}^{m-1}\) such that for any polynomial \(p\) of degree at most \(n\) and for \(k\in{\mathbb N}\) one has \[ m_k(p)=(t_{1,m}(\lambda_1))^k\int_{0}^{1}p(x)w(x)dx-\sum_{i=2}^{m}\delta_i^1\,p(t_{i,m}(\lambda_1)) \] and \[ m_k(p)=(t_{m,m}(\lambda_2))^k\int_{0}^{1}p(x)w(x)dx+\sum_{i=1}^{m-1}\delta_i^2\,p(t_{i,m}(\lambda_2)), \] where \(\lambda_1,\lambda_2\in(0,1)\) are defined by \(t_{1,m}(\lambda_1)=\lambda_1\) and \(t_{m,m}(\lambda_2)=\lambda_2\), respectively. Using this representation, the authors solve the problem of finding the maximum and minimum of \(m_k(p)\) among all polynomials \(p\) of degree at most \(n\) with \(p(x)\geq0\) for \(x\in[0,1]\) and \(m_0(p)=1\). Analogous results are given for \(n=2m-1\).