Weighted (0,1,3) interpolation on the zeros of Hermite polynomials (Q1310430)

At the suggestion of \textit{P. Turán}, \textit{J. Balázs} [Magyar. Tud. Akad., Mat. Fiz. Tvd. Oszt. Közl. 11, 305-338 (1961; Zbl 0129.049)], initiated the study of \((0,2)\)-interpolation polynomials, i.e. polynomials \(R_ n\) of degree \(\leq 3n\), satisfying \(R_ n(x_{k,n})= a_{k,n}\), \((wR_ n)''(x_{k,n})= b_{k,n}\), \(k=1,\dots,n\), where \(x_{k,n}\) are the roots of the ultraspherical polynomial \(P^{(\alpha)}_ n(x)\), \(\alpha>-1\) and \(w(x)= (1-x^ 2)^{{\alpha+1\over 2}}\), \(x\in [-1,1]\). The purpose of this paper is to extend the results from \((0,2)\)-interpolation theory to \((0,1,3)\)- interpolation which means to find a polynomial \(G_ n\) of degree \(\leq 3n\) such that \(G_ n(x_{k,n})= a_{k,n}\), \(G_ n(x_{k,n})= b_{k,n}\), \((\rho G_ n)'''(x_{k,n})= c_{k,n}\), \(k=1,\dots,n\), for \(x_{k,n}\) the roots of Hermite polynomial \(H_ n(x)\), \(a_{k,n}\), \(b_{k,n}\), \(c_{k,n}\) arbitrary numbers and \(\rho()= e^{- x^ 2}\), \(x\in R\). The paper is concerned with existence, unicity and convergence problems for these polynomials. The main result is the following: If \(f: R\to R\) is twice continuously differentiable and satisfies some growth conditions for \(| x|\to\infty\), then \[ e^{-\gamma x^ 2}| f(x)- G_ n(x)|= O(\log n)\cdot\omega(f',1/\sqrt n),\quad\text{for }\gamma> 3/2. \] {}.