Uniform convergence of higher order quasi Hermite-Fejér interpolation
Let \(X_ n = \{x_{kn} = \cos {k \pi \over n + 1},\;1 \leq k \leq n\}\) be the zeros of the Chebyshev polynomial of the second kind \(U_ n (x) = \sin (n + 1) \theta/ \sin \theta\). Let \(\overline X_ n = X_ n \cup \{- 1,1\}\). For any integer \(q \geq 0\), the author considers a quasi- Hermite-Fejér interpolation problem of higher order on the nodes \(\overline X_ n\). The polynomial \(Q_{n,q} (x,f)\) is of degree \(N = 2(q + 1) (n + 1) - 1\), interpolates \(f(x) \in C [- 1,1]\) on the nodes \(\overline X_ n\) and satisfies the following conditions as well: \[ \begin{aligned} Q^{(j)}_{nq} (x_{kn}, f) & = c_{j,k,n}, \quad 1 \leq k \leq n, \quad 1 \leq j \leq 2q + 1 \\ Q^{(j)}_{nq} (q,f) & = d_{jn}, \quad Q^{(j)}_{nk} (- 1,f) = g_{jn}, \quad 1 \leq j \leq q.\tag{1}\end{aligned} \] For \(q = 0\) the problem was considered by [\textit{R. B. Saxena} and \textit{K. B. Mathur} in Pac. J. Math. 20, 245-259 (1967; Zbl 0165.385)], and for \(q = 1\), the problem was treated by [\textit{A. Sharma} and \textit{J. Tzimbalario} in J. Approximation Theory 13, 431-442 (1975; Zbl 0302.41001)]. Here under suitable growth conditions on \(c_{j,k,n}\), \(d_{j,n}\) and \(g_{j,n}\), the author proves that \[ \| Q_{nq} (f) - f \| = O(1) \omega_ \varphi \left( f, {\ell n, n \over n} \right) \] where \(O\) is independent of \(f\) and \(n\), \(\varphi (x) = \sqrt {1 - x^ 2}\) and \(\omega_ \varphi (f,t)\) is the Ditzian and Totik modulus of \(f\). Two special cases of this result are stated and one of them is proved. One special case comprises special growth conditions on \(c_{j,k,n}\), \(d_{jn}\) and \(g_{jn}\) and the other case is the one when all these constants are taken to be zero.
scientific article; zbMATH DE number 640471
scientific article