The divergence of Lagrange interpolation for |x|α (2 < α < 4) at equidistant nodes

2006 ◽  
Vol 22 (2) ◽  
pp. 146-154
Author(s):  
Hui Su ◽  
Shusheng Xu ◽  
P. R. China
Keyword(s):  
Lagrange Interpolation ◽  
Equidistant Nodes

scholarly journals On Lagrange interpolation with equally spaced nodes

2000 ◽  
Vol 62 (3) ◽  
pp. 357-368 ◽  
Author(s):  
Michael Revers
Keyword(s):  
Rate Of Convergence ◽  
Lagrange Interpolation ◽  
End Points ◽  
Quantitative Version ◽  
Equidistant Nodes ◽  
Lagrange Interpolation Polynomials ◽  
Interpolation Polynomials ◽  
Exact Rate Of Convergence

A well-known result due to S.N. Bernstein is that 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|3 at equidistant nodes. Furthermore, we present the exact rate of convergence for the interpolatory parabolas at the point zero.

scholarly journals On Lagrange interpolation with equidistant nodes

1990 ◽  
Vol 42 (1) ◽  
pp. 81-89 ◽  
Author(s):  
Graeme J. Byrne ◽  
T.M. Mills ◽  
Simon J. Smith
Keyword(s):  
Lagrange Interpolation ◽  
Classical Result ◽  
Numerical Computations ◽  
Quantitative Version ◽  
Equidistant Nodes ◽  
Lagrange Interpolation Polynomials ◽  
Interpolation Polynomials

A quantitative version of a classical result of S.N. Bernstein concerning the divergence of Lagrange interpolation polynomials based on equidistant nodes is presented. The proof is motivated by the results of numerical computations.

scholarly journals On the divergence of Hermite-Fejér type interpolation with equidistant nodes

1994 ◽  
Vol 49 (1) ◽  
pp. 101-110 ◽  
Author(s):  
T.M. Mills ◽  
Simon J. Smith
Keyword(s):  
Lagrange Interpolation ◽  
Interpolation Polynomial ◽  
Simple Function ◽  
Convergence Properties ◽  
Lagrange Interpolation Polynomial ◽  
Type Interpolation ◽  
General Terms ◽  
Equidistant Nodes ◽  
Unique Polynomial

If f(x) is defined on [−1, 1], let H¯1 n(f, x) denote the Lagrange interpolation polynomial of degree n (or less) for f which agrees with f at the n+1 equally spaced points xk, n = −1 + (2k)/n (0 ≤ k ≤ n). A famous example due to S. Bernstein shows that even for the simple function h(x) = │x│, the sequence H¯1 n (h, x) diverges as n → ∞ for each x in 0 < │x│ < 1. A generalisation of Lagrange interpolation is the Hermite-Fejér interpolation polynomial H¯mn (f, x), which is the unique polynomial of degree no greater than m(n + 1) – 1 which satisfies (f, Xk, n) = δo, pf(xk, n) (0 ≤ p ≤ m − 1, 0 ≤ k ≤ n). In general terms, if m is an even number, the polynomials H¯mn(f, x) seem to possess better convergence properties than the H¯1 n (f, x). Nevertheless, D.L. Berman was able to show that for g(x) ≡ x, the sequence H¯2n(g, x) diverges as n → ∞ for each x in 0 < │x│. In this paper we extend Berman's result by showing that for any even m, H¯mn(g, x) diverges as n → ∞ for each x in 0 < │x│ < 1. Further, we are able to obtain an estimate for the error │H¯mn(g, x) – g(x)│.

scholarly journals On the Lebesgue function for lagrange interpolation with equidistant nodes

1992 ◽  
Vol 52 (1) ◽  
pp. 111-118 ◽  
Author(s):  
T. M. Mills ◽  
Simon J. Smith
Keyword(s):  
Asymptotic Expansion ◽  
Lower Bounds ◽  
Lagrange Interpolation ◽  
Upper And Lower Bounds ◽  
Lebesgue Function ◽  
Equidistant Nodes ◽  
Image Position

AbstractProperties of the Lebesgue function associated with interpolation at the equidistant nodes , are investigated. In particular, it is proved that the relative maxima of the Lebesgue function are strictly decreasing from the outside towards the middle of the interval [0, n], and upper and lower bounds, and an asymptotic expansion, are obtained for the smallest maximum when n is odd.

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