On Lagrange interpolation with equally spaced nodes

Abstract

A well-known result due to S.N. Bernstein is that the sequence of Lagrange interpolation polynomials for \x\ at equally spaced nodes in [-1,1] diverges everywhere, except at zero and the end-points. In this paper we present a quantitative version concerning the divergence behaviour of the Lagrange interpolants for |x| at equidistant nodes. Furthermore, we present the exact rate of convergence for the interpolatory parabolas at the point zero.

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; .…”

A note on Lagrange interpolation for |x|λ at equidistant nodes

“…Pn (R) This implies (8). Next using (4) again, we obtain liminf|sin(7r/3 fi n)| 1/(2n) ^ liminf|sin(7T^g n )| 1/(2?n(il)) = lim inf sin no n (/ Thus (9) follows.…”

“…was established for A = 1 by Byrne, Mills and Smith [3] and for A = 3 by the second author [8]. Recently, the first author [5] proved the conjecture posed in [8] that (1) holds for all A > 0, A ^ 2 , 4 , .…”

“…The classical result of Bernstein was revisited in the 1990s and 2000s. In particular, the rate of this divergence process was discussed in ( [3,5,6,8,10]). More precisely, the following nth root asymptotic relation for 0 < \x o \ < 1…”

The exact convergence rate at zero of Lagrange interpolation polynomial to |x|α

“…As we know, the larger the α is, the more smoothness the |x| α has, so Revers conjecture is very valuable for α ≥ 1.When α is an odd integer, there is a good result .For example when α = 3, such a result is given in [5]:…”