Pál-type Hermite interpolation on infinite interval
The paper deals with the so-called Pál-type Hermite interpolation . It is well known that for each even natural number \(n\) and arbitrary numbers \((\alpha_j^{(n)})^n_{j = 1}\), \((\beta_j^{(n)})^{n - 1}_{j = 1}\) there exists a uniquely determined polynomial \(P_n\) of degree at most \(2n - 1\) such that \[ \begin{multlined} P_n \bigl( x_j^{(n)} \bigr) = \alpha_j^{(n)},\;j = 1, \ldots, n, \;P_n' \bigl( y_j^{(n)} \bigr) = \beta_j^{(n)},\;j = 1, \ldots, n - 1,\\ P_n (0) = \sum^n_{j = 1} \alpha_j^{(n)} l_j^{(n)} (0)^2, \end{multlined} \tag{*} \] where the numbers \((x_j^{(n)})^n_{j = 1}\) and \((y_j^{(n)})^{n - 1}_{j = 1}\) are the zeros of the \(n\)-th Hermite polynomial \(H_n\) and its first derivative \(H_n'\), and the polynomials \((l_j^{(n)})^n_{j = 1}\) are the fundamental polynomials of Lagrange interpolation with respect to \((x_j^{(n)})^n_{j = 1}\). In the present paper an explicit expression for the solution \(P_n\) is given in terms of fundamental polynomials of first and second kind. Moreover, for a suitable class of continuously differentiable functions \(f : \mathbb{R} \to \mathbb{R}\) the term \(e^{- \nu x^2} |P_n (f,x) - f(x) |\) is estimated on the whole real line, where \(\nu > 1\) and the polynomials \(P_n (f, \cdot)\) are determined by \((*)\) taking \((\alpha_j^{ (n)})^n_{j = 1} = (f(x_j^{(n)}))^n_{j = 1}\) and \((\beta_j^{(n)})^{n - 1}_{j = 1} = (f' (y_j^{(n)}))^{n - 1}_{j = 1}\).
scientific article; zbMATH DE number 785117