scholarly journals The Lebesgue Function and Lebesgue Constant of Lagrange Interpolation for Erdoős Weights

1998 ◽  
Vol 94 (2) ◽  
pp. 235-262 ◽  
Author(s):  
S.B. Damelin
Keyword(s):  
Lagrange Interpolation ◽  
Lebesgue Constant ◽  
Lebesgue Function

scholarly journals Multivariate Lagrange Interpolation at Sinc Points Error Estimation and Lebesgue Constant

2016 ◽  
Vol 8 (4) ◽  
pp. 118 ◽  
Author(s):  
Maha Youssef ◽  
Hany A. El-Sharkawy ◽  
Gerd Baumann
Keyword(s):  
Uniform Convergence ◽  
Error Estimation ◽  
Upper Bound ◽  
Lagrange Interpolation ◽  
Lebesgue Constant ◽  
Explicit Construction ◽  
Interpolation Operator ◽  
Lebesgue Constants ◽  
Set Of Points

This paper gives an explicit construction of multivariate Lagrange interpolation at Sinc points. A nested operator formula for Lagrange interpolation over an $m$-dimensional region is introduced. For the nested Lagrange interpolation, a proof of the upper bound of the error is given showing that the error has an exponentially decaying behavior. For the uniform convergence the growth of the associated norms of the interpolation operator, i.e., the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature $O((log n)^m)$. We compare the obtained Lebesgue constant bound with other well known bounds for Lebesgue constants using different set of points.

scholarly journals The Lebesgue function for generalized Hermite-Fejér interpolation on the Chebyshev nodes

2000 ◽  
Vol 42 (1) ◽  
pp. 98-109 ◽  
Author(s):  
Graeme J. Byrne ◽  
T. M. Mills ◽  
Simon J. Smith
Keyword(s):  
Asymptotic Expansion ◽  
Chebyshev Polynomial ◽  
Lebesgue Constant ◽  
Local Maximum ◽  
Lebesgue Function ◽  
Chebyshev Nodes ◽  
Short Survey ◽  
Convergence Results ◽  
Limiting Behaviour

AbstractThis paper presents a short survey of convergence results and properties of the Lebesgue function λm,n(x) for(0, 1, …, m)Hermite-Fejér interpolation based on the zeros of the nth Chebyshev polynomial of the first kind. The limiting behaviour as n → ∞ of the Lebesgue constant Λm,n = max{λm,n(x): −1 ≤ x ≤ 1} for even m is then studied, and new results are obtained for the asymptotic expansion of Λm,n. Finally, graphical evidence is provided of an interesting and unexpected pattern in the distribution of the local maximum values of λm,n(x) if m ≥ 2 is even.

scholarly journals On the Lebesgue function of weighted Lagrange interpolation. II

1998 ◽  
Vol 65 (2) ◽  
pp. 145-162 ◽  
Author(s):  
P. Vértesi
Keyword(s):  
Lower Bounds ◽  
Lagrange Interpolation ◽  
Lebesgue Function

AbstractThe aim of this paper is to continue our investigation of the Lebesgue function of weighted Lagrange interpolation by considering Erdős weights on ℝ and weights on [−1, 1]. The main results give lower bounds for the Lebesgue function on large subsets of the relevant domains.

scholarly journals The Lebesgue function for Hermite-Fejér interpolation on the extended Chebyshev nodes

2002 ◽  
Vol 66 (1) ◽  
pp. 151-162
Author(s):  
Simon J. Smith
Keyword(s):  
Chebyshev Polynomial ◽  
Minimum Degree ◽  
Convergence Property ◽  
Lebesgue Constant ◽  
Interpolation Polynomial ◽  
Lebesgue Function ◽  
Surprising Result ◽  
End Points ◽  
Chebyshev Nodes ◽  
Interpolation Polynomials

Given f ∈ C[−1, 1] and n point (nodes) in [−1, 1], the Hermite-Fejér interpolation polynomial is the polynomial of minimum degree which agrees with f and has zero derivative at each of the nodes. In 1916, L. Fejér showed that if the nodes are chosen to be zeros of Tn (x), the nth Chebyshev polynomial of the first kind, then the interpolation polynomials converge to f uniformly as n → ∞. Later, D.L. Berman demonstrated the rather surprising result that this convergence property no longer holds true if the Chebyshev nodes are extended by the inclusion of the end points −1 and 1 in the interpolation process. The aim of this paper is to discuss the Lebesgue function and Lebesgue constant for Hermite-Fejér interpolation on the extended Chebyshev nodes. In particular, it is shown that the inclusion of the two endpoints causes the Lebesgue function to change markedly, from being identically equal to 1 for the Chebyshev nodes, to having the form 2n2(1 − x2)(Tn (x))2 + O (1) for the extended Chebyshev nodes.

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