Hermite-Fejér interpolation of higher order with varying weights
Let \(X=(x_{i,n}=x_i \in I=[-1,1]: i=1, \cdots, n; \;n \in \mathbb N)\) be an infinite matrix where each row is a set of distinct points in \(I\). For \(f \in C(I)\) the unique Hermite-Féjer interpolation polynomial \(H_{nq}(F,X,x)\) of order \(\leq nq-1\) (\(q \geq 1\) a fixed integer) is defined by \[ H^{(\ell)}_{nq}(f,X,x_{kn})=\delta_{0 \ell} f(x_{kn}), k=1, \cdots, n; \ell =1, \cdots, q-1. \] This implies that \[ H_{nq}(F,X,x) = \sum_{k=1}^n f(x_{kn}) h_{k n q}(x), \quad n = 1, 2, 3, \cdots, \] with \(h_{k n q}(x)\) polynomial of degree exactly \(nq-1\) such that \[ h^{(\ell)}_{k n q}(x_{j n})= \delta_{0 \ell} \delta_{k j}, 1 \leq k, j \leq n, 0 \leq \ell \leq q-1. \] An explicit form of \(h^{(\ell)}_{k n q}\) is \[ h^{(\ell)}_{k n q}(x) = \ell^q_{kn}(x) \sum_{i=0}^{q-1} e_{iknq}(x-x_{kn})^i, 1 \leq k \leq n, \] where \(\ell_{kn}\) are the Lagrange fundamental polynomials of degree exactly \(n-1\). When the interpolation nodes are the \(n\)th Jacobi polynomials \(P^{\alpha, \beta}(x), \alpha, \beta > -1\), then some properties of \(H_{nq}\) were investigated by \textit{R. Sakai} and \textit{P. Vértesi} [Stud. Sci. Math. Hung. 28, No. 1-2, 87--97 (1993; Zbl 0802.41006) and Stud. Sci. Math. Hung. 28, No. 3-4, 379--386 (1993; Zbl 0832.41003)]. In particular, they obtained precise estimates for the coefficients \(e_{iknq}\). In the above mentioned Jacobi case the coefficients can be estimated by the well known differential equation. However this differential equation is no longer available for the more general case of generalized Jacobi polynomials considered in the paper under review. To overcome this difficulty, the authors obtain precise estimates for the coefficients based on properties of zeros of these generalized Jacobi polynomials.
scientific article; zbMATH DE number 5121505