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2000•Journal of Liaoning Normal UniversityRequires access

Linear Combination of Lagrange Interpolation Polynomial

Xuegang Yuan

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Abstract

Because the Lagrange interpolation polynomial does not converge uniformly for an arbitrary continu ous function ,in this paper,the interpolated functions are linearly combi ned in Lagrange interpolation polynomial.An operator A n,r(f;x) is cons tructed based on the zeros of (1-x)W n(x) as the interpolation nodes.It con verges uniformly to the concerned arbitrary continuous,and derivable function f(x)∈C-l [- 1,1],(0≤l≤r) and the convergence order is ｜A n,r(f;x)-f (x)｜=OE n(f)+ 1n-lω(f- (l),1n)+1n- l+1,and the convergence order is the best.(Where r is an odd natural number.)

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What this paper is about

Because the Lagrange interpolation polynomial does not converge uniformly for an arbitrary continu ous function ,in this paper,the interpolated functions are linearly combi ned in Lagrange interpolation polynomial.An operator A n,r(f;x) is cons tructed based on the zeros of (1-x)W n(x) as the interpolation nodes.It con verges uniformly to the concerned arbitrary continuous,and derivable function f(x)∈C-l [- 1,1],(0≤l≤r) and the convergence order is ｜A n,r(f;x)-f (x)｜=OE n(f)+ 1n-lω(f- (l),1n)+1n- l+1,and the convergence order is the best.(Where r is an odd natural number.)

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Available abstract

Because the Lagrange interpolation polynomial does not converge uniformly for an arbitrary continu ous function ,in this paper,the interpolated functions are linearly combi ned in Lagrange interpolation polynomial.An operator A n,r(f;x) is cons tructed based on the zeros of (1-x)W n(x) as the interpolation nodes.It con verges uniformly to the concerned arbitrary continuous,and derivable function f(x)∈C-l [- 1,1],(0≤l≤r) and the convergence order is ｜A n,r(f;x)-f (x)｜=OE n(f)+ 1n-lω(f- (l),1n)+1n- l+1,and the convergence order is the best.(Where r is an odd natural number.)

Key concepts: Lagrange polynomial, Mathematics, Interpolation (computer graphics), Polynomial interpolation, Polynomial, Convergence (economics), Function (biology), Order (exchange)

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