Hermite-Fejér interpolations of higher order. I
Let \(X=\{x_{kn}\}\subset [-1,1]\), with \(-1\leq x_{nn}<...<x_{1n}\leq 1\), \((n=1,2,...)\) be an infinite triangular matrix. Let \(H_{nm}(f,X,x)\) be the unique polynomial of degree mn-1 for \(f\in C[-1,1]\) which interpolates f on the nodes \(\{x_{kn}\}^ n_{n=1}\), and whose derivatives of order up to m-1 vanish at these nodes. These polynomials are called Hermité-Féjer interpolations of higher order. A usual question regarding these polynomials is to find matrices X and values of m such that \((1)\quad \lim_{n\to \infty}\| H_{mn}(f,X,x)-f(x)\| =0\) for every \(f\in C\). Faber was the first to raise this issue for \(m=1\) and to show that for any X, there is an \(f\in C\) for which (1) does not hold. For \(m=2\), Szegö proved that if \(X=X^{(\alpha,\beta)}\), i.e., the nodes are the zeros of the nth Jacobi polynomials \(P_ n^{(\alpha,\beta)}\), \(\alpha,\beta >-1\), then (1) holds for any \(f\in C\) whenever \(-1<\alpha <\beta <0\). He showed that if max(\(\alpha\),\(\beta\))\(\geq 0\), then there is an \(f\in C\) such that (1) does not hold. For \(m=4\), the author proved [Stud. Sci. Math. Hung. 22, No.1-4, 239-245 (1987; Zbl 0659.41010)] that if \(-1<\alpha <\beta <-1/4\) and \(| \alpha -\beta | <1/2\), then (1) holds for any \(f\in C\). This result was sharp in the sense that if max(\(\alpha\),\(\beta\))\(\geq - 1/4\) or \(| \alpha -\beta | \geq 1/2\), then (1) is not true for a certain \(f\in C\). Here the author investigates the convergence-divergence behaviour of \(H_{mn}^{(\alpha,\beta)}\), \(\alpha\),\(\beta\geq -1\) on any interval [c,d]\(\subseteq [-1,1]\) for fixed m, and give a brief account of some recent results of R. Sakai and of R. Shibata.
scientific article; zbMATH DE number 4153126