Exact order of the Lebesgue constants for bivariate Lagrange interpolation at certain node-systems

The authors investigate two sets of nodes for bivariate Lagrange interpolation in [−1, 1] 2 and prove that the order of the corresponding Lebesgue constants is (log n ) 2 .

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Multivariate Lagrange Interpolation at Sinc Points Error Estimation and Lebesgue Constant

Journal of Mathematics Research ◽

10.5539/jmr.v8n4p118 ◽

2016 ◽

Vol 8 (4) ◽

pp. 118 ◽

Cited By ~ 2

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.

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Influence of the choice of Lagrange interpolation nodes on the exact and approximate values of the Lebesgue constants

Siberian Mathematical Journal ◽

10.1134/s0037446614060184 ◽

2014 ◽

Vol 55 (6) ◽

pp. 1144-1160 ◽

Cited By ~ 3

Author(s):

I. A. Shakirov

Keyword(s):

Lagrange Interpolation ◽

Lebesgue Constants ◽

Interpolation Nodes

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A link between Lebesgue constants and Hermite-Fejér interpolation

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700003075 ◽

1986 ◽

Vol 33 (2) ◽

pp. 207-218 ◽

Cited By ~ 1

Author(s):

S. J. Goodenough

Keyword(s):

Rate Of Convergence ◽

Lagrange Interpolation ◽

Historical Perspective ◽

Lebesgue Constant ◽

The Other ◽

Lebesgue Constants ◽

Close Link ◽

The One ◽

Image Position ◽

Interpolation Polynomials

A review of the development of estimates for Lebesgue constants associated with Lagrange interpolation on the one hand, and estimates for the rate of convergence of Hermite-Fejér interpolation on the other hand, provides a historical perspective for the following surprising, close link between these apparently diverse concepts. Denoting by Λn (T) the Lebesgue constant of order n and by Δn (T) the maximum interpolation error for functions of class Lip 1 by Hexmite-Fejér interpolation polynomials of degree not exceeding 2n − 1, based on the zeros of the Chebyshev polynomial of first kind, we discover that, for even values of n, Λn(T) = n Δn(T).

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Exact order of the lebesgue constants of hyperbolic partial sums of multiple fourier series

Mathematical Notes ◽

10.1007/bf01156675 ◽

1986 ◽

Vol 39 (5) ◽

pp. 369-374 ◽

Cited By ~ 2

Author(s):

I. R. Liflyand

Keyword(s):

Fourier Series ◽

Multiple Fourier Series ◽

Partial Sums ◽

Exact Order ◽

Lebesgue Constants

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A Note on the Accuracy of Normal Approximation of Random Quantities

Calcutta Statistical Association Bulletin ◽

10.1177/00080683211013510 ◽

2021 ◽

Vol 73 (1) ◽

pp. 62-67

Author(s):

Ibrahim A. Ahmad ◽

A. R. Mugdadi

Keyword(s):

Sample Size ◽

Order Statistic ◽

Normal Approximation ◽

Order Approximation ◽

Random Variables ◽

Random Variable ◽

Limiting Distribution ◽

Exact Order ◽

Fixed Sample ◽

Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

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