Hermite interpolation at the zeros of certain Freud-type orthogonal polynomials
Assume that \[ A:=\{x_{kn},\;k=1,\ldots,n,\;n=1,2\ldots\}, \tag{1.1} \] \[ B:=\{y_{kn},\;k=1,\ldots,n,\;n=1,2\ldots\}, \tag{1.2} \] be triangular schemes of real numbers and \(f: \mathbb{R}\to\mathbb{R}\). The Hermite interpolation polynomial \(H_ n(f,x):=H_ n(A,B,f,x)\) is the unique polynomial of degree at most \(2n-1\) which satisfies (1.3) \(H_ n(f,x_{kn})=f(x_{kn})\), \(H_ n'(f,x_{kn})=y_{kn}\), \(k=1,\ldots,n\), \(n=1,2\ldots\). When all the \(y_{kn}\)'s are equal to zero, \(H_ n(A,B,f)\) is called the Hermite-Fejér interpolation polynomial. In 1930, L. Fejér proved the following theorem [cf. Constructive function Theory by \textit{I. P. Natanson} (1965; Zbl 0178.397)]. Theorem 1.1 Let \(f\) be continuous on \([-1,1]\), \(x_{kn}=\cos\bigl({2k-1\over 2n}\pi\bigr)\), \(k=1,\ldots,n\), \(n=1,2,\ldots\) and \[ \lim_{n\to\infty}\max_{1\leq k\leq n}\left\{{\log n\over n}\sqrt{1-x^ 2_{kn}}| y_{kn}|\right\}=0. \] Then the sequence \(\bigl\{H_ n(A,B,f)\bigr\}\) converges to \(f\) uniformly on \([- 1,1]\) as \(n\to\infty\). \textit{R. Bojanic} [Proc. Conf. construct Theory Functions (Approximation Theory) 1969, 69-76 (1972; Zbl 0259.41001)] proved the following theorem. Theorem 1.2 Let \(f: [-1,1]\to\mathbb{R}\) be continuous on \([-1,1]\) and for \(\delta>0\), we put \[ \omega(f,\delta):=\max\bigl\{| f(x)-f(t)|,\;| x- t|\leq\delta,\;x,t\in[-1,1]\bigr\}. \tag{1.4} \] Let \(\{x_{kn}\}\) be as in Theorem 1.1 and \(y_{kn}\)'s be all equal to zero. Then for \(x\in[- 1,1]\), \[ | H_ n(f,x)-f(x)|\leq{c\over n}\sum^ n_{k=1}\omega\left(f,{1\over k}\right),\quad n=1,2,\ldots \tag{1.5} \] where \(c\) is an absolute constant. ``With the aid of recent research in the theory of orthogonal polynomials, it has now become possible to obtain the analogues of Theorems 1.1 and 1.2, when the \(x_{kn}\)'s are the zeros of polynomials orthogonal on the whole real line with respect to a weight function of the form \(w^ 2_ Q(x):=\exp\bigl(-2Q(x)\bigr)\) satisfying various technical conditions. In this paper, the authors assume only that \(w^ p_ Qf\in C_ 0(R)\) for some \(p\), \(0<p<2\) and show that under certain conditions on \(B\), \(H_ n(A,B,f)\) converges uniformly on compact subsets of \(R\) when \(x_{kn}\)'s are as described. The authors also obtain an estimate analogous to (1.5) for the rate of convergence, using a modulus of continuity which is natural for weighted polynomial approximation''.
scientific article; zbMATH DE number 149633