A generalization of Pál interpolation. III: The Hermite case
Let \( -\infty <x_1< \dots < x_n <\infty\) be distinct nodes, \(\omega (x):=\prod_{k=1}^n (x-x_k),\) and let \(x'_k\), \(x_k < x'_k < x_{k+1}\), be the roots of the derivative \(\omega ' (x)\). Taking \( \omega (x):= H_n (x)\) -- the \(n\)-th Hermite polynomial, and denoting by \(x^*_1, \dots , x^* _{n^*}\) the real roots of \( \Omega (x):= - n H_n (x)+ x H'_n (x)\), one looks for a polynomial \(R_n (x)\), of minimal degree, such that \( R_n (x_k) =y_k\), \(k=1,\dots ,n\), and \(R'_n (x'_k)= y^* _k\), \(k=1,\dots ,n^*\), where \(y_1,\dots ,y_n\) and \(y^* _1,\dots ,y^* _{n^*}\), are arbitrarily given numbers. Using some results from his previous paper [part II, Acta Math. Hung. 74, No. 1-2, 19-29 (1997; see the preceding review)], the author gives an explicit formula for the interpolation polynomial \(R_n (x)\) (of degree \(n+(n-2)-1\)) and proves a convergence result: \(|f(x)-R_n(x)|\leq O(1) \omega(f',1/\sqrt n),\) for \(f:{\mathbb{R}}\to \mathbb{R}\) continuously differentiable and satisfying the growth conditions \( x^{2r} f(x) \exp(-x^2/2) \to 0\) and \( f'(x) \exp(-x^2/2) \to 0\), for \(|x|\to \infty\). In this case \( y_k =f(x_k)\) and \( y'_k= f'(x_k)\).
scientific article; zbMATH DE number 1288363