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 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.

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.

scholarly journals Lebesgue Constants and Optimal Node Systems via Symbolic Computations

10.29007/89cm ◽  
2018 ◽  
Author(s):  
Robert Vajda
Keyword(s):  
Polynomial Interpolation ◽  
Classical Method ◽  
Lebesgue Constant ◽  
Quantifier Elimination ◽  
Continuous Functions ◽  
Interpolation Polynomial ◽  
Lebesgue Function ◽  
Lebesgue Constants ◽  
The Best Approximation ◽  
Optimal Node

Polynomial interpolation is a classical method to approximatecontinuous functions by polynomials. To measure the correctness of theapproximation, Lebesgue constants are introduced. For a given node system $X^{(n+1)}=\{x_1<\ldots<x_{n+1}\}\, (x_j\in [a,b])$, the Lebesgue function $\lambda_n(x)$ is the sum of the modulus of the Lagrange basis polynomials built on $X^{(n+1)}$. The Lebesgue constant $\Lambda_n$ assigned to the function $\lambda_n(x)$ is its maximum over $[a,b]$. The Lebesgue constant bounds the interpolation error, i.e., the interpolation polynomial is at most $(1+\Lambda_n)$ times worse then the best approximation.The minimum of the $\Lambda_n$'s for fixed $n$ and interval $[a,b]$ is called the optimal Lebesgue constant $\Lambda_n^*$.For specific interpolation node systems such as the equidistant system, numerical results for the Lebesgue constants $\Lambda_n$ and their asymptoticbehavior are known \cite{3,7}. However, to give explicit symbolic expression for the minimal Lebesgue constant $\Lambda_n^*$ is computationally difficult. In this work, motivated by Rack \cite{5,6}, we are interested for expressing the minimalLebesgue constants symbolically on $[-1,1]$ and we are also looking for thecharacterization of the those node systems which realize theminimal Lebesgue constants. We exploited the equioscillation property of the Lebesgue function \cite{4} andused quantifier elimination and Groebner Basis as tools \cite{1,2}. Most of the computation is done in Mathematica \cite{8}.

Unexpectedness

2019 ◽  
Vol 6 (1) ◽  
Author(s):  
John-Carlos Perea ◽  
Jacob E. Perea
Keyword(s):  
American Indian ◽  
San Francisco ◽  
Interdisciplinary Research ◽  
Ethnic Studies ◽  
College Of Education ◽  
Present Moment ◽  
State University ◽  
Research Projects ◽  
Short Survey ◽  
Wide Range

The concepts of expectation, anomaly, and unexpectedness that Philip J. Deloria developed in Indians in Unexpected Places (2004) have shaped a wide range of interdisciplinary research projects. In the process, those terms have changed the ways it is possible to think about American Indian representation, cosmopolitanism, and agency. This article revisits my own work in this area and provides a short survey of related scholarship in order to reassess the concept of unexpectedness in the present moment and to consider the ways my deployment of it might change in order to better meet the needs of my students. To begin a process of engaging intergenerational perspectives on this subject, the article concludes with an interview with Dr. Jacob E. Perea, dean emeritus of the Graduate College of Education at San Francisco State University and a veteran of the 1969 student strikes that founded the College of Ethnic Studies at San Francisco State University.

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