One-sided convergence conditions of Lagrange interpolation based on Jacobi-type weights (Q1878573)

The authors state a one-side convergence theorem for Lagrange interpolation. This result says that if \[ -1<\gamma=\max (\alpha,\beta)<\frac 1 2 \] is fixed, \(f\in C(I)\) and \[ \delta(f;t)=\begin{cases} o\big(| \log t| ^{-1}\big), & \text{if}\;-1<\gamma\leq \dfrac 1 2 \\ o\big(t^{\gamma+\frac 1 2}\big), & \text{if}\;-\dfrac 1 2 <\gamma <\dfrac 1 2, \end{cases} \] then \(\lim_{n\to\infty}\| L_n^{(\alpha,\beta)} (f)-f\| =0\). Here \(\| f\| \) is the usual supremum norm of \(f\) on \(I\), moreover \(\delta (f;t)\) denotes the so called one-sided modulus of continuity of \(f\). A crucial point for the investigations of Lagrange interpolation is the knowledge of precise estimates for the first derivative at the nodes of the polynomials whose zeros are the interpolation knots. In the Jacobi case the derivative can be easily estimated by the well known differential equation. In the case of generalized Jacobi polynomials the authors use another way to estimate the derivatives. The authors extend this result to the Lagrange interpolation based on generalized Jacobi zeros. Also, they prove that in a sense the results are best possible.