Turán type problems on mean convergence. II: Hermite-Fejér type interpolations
This is the second part of the precedent paper. The notations are the same but this time \(m\in \{2,4,\dots\}\) are a fixed even number. For \(m= 2\), \(H_{n,2}\) is the classical Hermite-Fejér interpolation polynomials and, as it is well known, \(H_{n,2}(f)\) converges uniformly to \(f\), for every \(f\in C\). This result was extended to an arbitrary even number \(m\) by \textit{R. Sakai} and \textit{P. Vértesi} [Stud. Sci. Math. Hung. 28, No. 1-2, 87-97 (1993; Zbl 0802.41006)]. In this case there exist weights \(u\) and \(w\) and a \(w\)-regular interpolatory matrix \(X\) such that for every \(p\), \(0< p<\infty\), \[ \varlimsup_{n\to \infty} \| f(x)- H_{n,m} (f, X, x)\|_{p, u}\| \omega^ s_ n(X, x)\|_{1,w}\leq \lim_{n\to \infty} {\varphi_ n\over n} \| \chi_{\Delta_ n} \omega^ s_ n(X, x)\|_{p,u}, \] for every \(f\in C\). Here \(\{\varphi_ n\}\) is an arbitrary sequence with \[ 0< \varphi_ 1< \varphi_ 2<\cdots,\;\lim_{n\to \infty} \varphi_ n= \infty,\;\Delta_ n= [-1, x_{n,n}]\cup [x_{1,n}, 1] \] and \(\chi_{\Delta_ n}\) is the characteristic function of \(\Delta_ n\).
convergence of interpolation polynomials
scientific article; zbMATH DE number 638252