On weighted \((0,2)\)-interpolation on infinite interval \((-\infty,+\infty)\) (Q2714172)

The paper investigates the Turán-type interpolation problem: to define the polynomial \(R_n\) of lowest possible degree satisfying the conditions NEWLINE\[NEWLINER_n(x_{i,n})=y_{i,n}, \quad (wR_n)^{''}(x_{i,n})=y_{i,n}^{''}, \quad i=1,\dots,n,NEWLINE\]NEWLINE where \(w \in C^2(a,b)\) is a weighted function and \(x_{i,n}\) are some interpolation basic points. The authors prove that taking the roots of the \(n\)-th Hermite polynomal \(H_n(x)\) as nodes, for each positive integer \(n\) there exists such a unique weighted \((0,2)\)-interpolatory polynomial \(R_n\) of degree \(\leq 2n\) satisfying the additional condition \(R_n(0)=y_{0,n}\) if \(n\) is even, and \(R_n^{'}(0)=y_{0,n}^{'}\) if \(n\) is odd.