Yudin-Hermite extremal problems for polynomials
Let \(\sigma(x):[-1,1]\to\mathbb{R}\) be a bounded nondecreasing function with infinitly many growth points, \(n,k\in\mathbb{Z}_+\), let \(\mathcal{P}_n\) be the set of algebraic polynomials of degree at most \(n\) with real valued coefficients, let \(\mathcal{P}\) be the set of all algebraic polynomials, let \[ \mu_k(p)=\int_{-1}^{1}x^kp(x)d\sigma(x) \] be the \(k\)th moment of a polynomials \(p\in\mathcal{P}\) on \([-1,1]\). Next, let consider three classes of polynomials defined by boundary and orthogonality conditions as follows. Let \(n\in\mathbb{N}\), \(m,k,l\in\mathbb{Z}_+\), \begin{align*} \mathcal{H}_n:=&\mathcal{H}_n(m,k,l)= \left\{p\in\mathcal{P}_n: p^{(i)}(1)=0, i=0,\dots,k-2,\right.\\ &\left. (-1)^kp^{(k-1)}(1)\geq 0;\quad p^{(i)}(-1)=0, i=0,\dots,l-2,\right.\\ &\left. (-1)^{m+1}p^{(l-1)}(-1)\geq 0 \right\}, \end{align*} \begin{align*} \mathcal{K}_n:=&\mathcal{K}_n(k,l)= \left\{p\in\mathcal{P}_n: p^{(i)}(1)=0, i=0,\dots,k-2,\right.\\ &\left. (-1)^kp^{(k)}(1)\geq 0;\quad p^{(i)}(-1)=0, i=0,\dots,l-1\right\}, \end{align*} \[\mathcal{O}_n:=\mathcal{O}_n(k,l)= \left\{p\in\mathcal{P}_n: \mu_j(p)=0, j=0,\dots,m-1, \mu_m(p)\geq 0\right\}.\] In the paper under review the author solved the following extremal problems for polynomials named here ''Yudin-Hermite problems'' Yudin-Hermite problem I: Evaluate \begin{align*} A_n(\sigma,m,k,l)=&\sup \left\{\lambda\in[-1,1]: (-1)^{m+1}p(x)\geq 0,\right.\\ &\left. -1\leq x\leq \lambda, p\in\mathcal{H}_n\cap\mathcal{O}_n\right\}. \end{align*} Yudin-Hermite problem II: Evaluate \begin{align*} B_n(\sigma,m,k,l)=&\sup \left\{\lambda\in[-1,1]: (-1)^{m+1}p(x)\geq 0,\right.\\ &\left. -1\leq x\leq \lambda, p\in\mathcal{K}_n\cap\mathcal{O}_n\right\}. \end{align*} The solutions are in terms of the system of polynomials \(\left\{p^{(\alpha,\beta)}_n(x)\right\}_{n=0}^{\infty}\), \(\alpha,\beta>-1\), orthogonal on \([-1,1]\) with respect to the measure \((1-x)^{\alpha}(1+x)^{\beta}d\sigma(x)\) and their zeros.
scientific article; zbMATH DE number 7455559