On Lagrange interpolation with equidistant nodes

Byrne, Graeme; Mills, Terry; Smith, Simon J.

doi:10.1017/s0004972700028161

Cited by 18 publications

(16 citation statements)

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“…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; .…”

Section: Introductionmentioning

confidence: 99%

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

Ganzburg

2002

Journal of Approximation Theory

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Section: Introductionmentioning

confidence: 99%

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

Ganzburg

2002

Journal of Approximation Theory

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“…Much attention has been devoted to the approximation of x j j and x j j a by polynomials. For a survey of contributions to these topics, see Bernstein [2], Brutman and Passow [3], Byrne, Mills and Smith [4], Li and Saff [5], Revers [8], Runck [10] and Varga and Carpenter [11].…”

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

Revers

2000

Archiv der Mathematik

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“…Thus Lagrange interpolation on equidistant nodes diverges for a simple function such as h (x) . A quantitative version of Bernstein's result was developed by Byrne, Mills and Smith [3], who showed that if 0 < |x| < 1, then…”

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

Mills

Smith

1994

Bull. Austral. Math. Soc.

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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)│.

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On Lagrange interpolation with equidistant nodes

Abstract: 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.

Cited by 18 publications

References 3 publications

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

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

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

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