On Lagrange interpolatory parabolas to $\left | x\right | ^{\alpha }$ at equally spaced nodes

Abstract

In this paper we present a generalized quantitative version of a result due to D. L. Berman concerning the exact convergence rate at zero of Lagrange interpolation polynomials to x j j a based on equally spaced nodes in À1Y 1 . The estimates obtained turn out to be best possible.

On Lagrange interpolation with equally spaced nodes

“…On the other hand, turning to Theorem 4 combined with the results of Berman and Lozinskii, no choice is left but to believe that the exact rate of convergence at the point zero for the interpolants of |x| Q is given by O{n~a). This, if true, would match perfectly with the following result established in [11]: THEOREM 6 . Let m € N, n = 2m -1 and 0 ^ a ^ 1.…”

“…There is a wide range of literature around Bernstein's classical divergence result. For example, see [2,3,7,9,10,11,12,14]. An extension of Bernstein's result is given in [10]: [2] Theorem 1 informs us that the divergence behaviour is rather general and does not depend on the special characteristics of \x\.…”

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

“…Approximation properties of the Lagrange interpolation polynomials to f l have attracted much attention in the 1990s and 2000s [7,8,13,[17][18][19][20][21]. In particular, Revers [17] proved that for N ¼ 2; 4; .…”