scholarly journals Classical Lagrange Interpolation Based on General Nodal Systems at Perturbed Roots of Unity

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 498
Author(s):  
Elías Berriochoa ◽  
Alicia Cachafeiro ◽  
Alberto Castejón ◽  
José Manuel García-Amor
Keyword(s):  
Rate Of Convergence ◽  
Unit Circle ◽  
Lagrange Interpolation ◽  
Roots Of Unity ◽  
Smooth Functions ◽  
Numerical Examples ◽  
Practical Applications ◽  
Bounded Interval ◽  
Interesting Approach ◽  
Nodal System

The aim of this paper is to study the Lagrange interpolation on the unit circle taking only into account the separation properties of the nodal points. The novelty of this paper is that we do not consider nodal systems connected with orthogonal or paraorthogonal polynomials, which is an interesting approach because in practical applications this connection may not exist. A detailed study of the properties satisfied by the nodal system and the corresponding nodal polynomial is presented. We obtain the relevant results of the convergence related to the process for continuous smooth functions as well as the rate of convergence. Analogous results for interpolation on the bounded interval are deduced and finally some numerical examples are presented.

scholarly journals A note on the rate of convergence for Chebyshev-Lobatto and Radau systems

2016 ◽  
Vol 14 (1) ◽  
pp. 156-166
Author(s):  
Elías Berriochoa ◽  
Alicia Cachafeiro ◽  
Jaime Díaz ◽  
Eduardo Martínez
Keyword(s):  
Rate Of Convergence ◽  
Lagrange Interpolation ◽  
Hermite Interpolation ◽  
Smooth Functions ◽  
Nodal Points

AbstractThis paper is devoted to Hermite interpolation with Chebyshev-Lobatto and Chebyshev-Radau nodal points. The aim of this piece of work is to establish the rate of convergence for some types of smooth functions. Although the rate of convergence is similar to that of Lagrange interpolation, taking into account the asymptotic constants that we obtain, the use of this method is justified and it is very suitable when we dispose of the appropriate information.

scholarly journals Rate of Convergence of Hermite-Fejér Interpolation on the Unit Circle

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
E. Berriochoa ◽  
A. Cachafeiro ◽  
J. Díaz ◽  
E. Martínez-Brey
Keyword(s):  
Rate Of Convergence ◽  
Unit Circle ◽  
Complex Number ◽  
Analytic Functions ◽  
Order Of Convergence ◽  
Supremum Norm ◽  
Laurent Polynomials ◽  
Nodal System ◽  
Norm Of The Error

The paper deals with the order of convergence of the Laurent polynomials of Hermite-Fejér interpolation on the unit circle with nodal system, thenroots of a complex number with modulus one. The supremum norm of the error of interpolation is obtained for analytic functions as well as the corresponding asymptotic constants.

scholarly journals On Lagrange interpolation with equally spaced nodes

2000 ◽  
Vol 62 (3) ◽  
pp. 357-368 ◽  
Author(s):  
Michael Revers
Keyword(s):  
Rate Of Convergence ◽  
Lagrange Interpolation ◽  
End Points ◽  
Quantitative Version ◽  
Equidistant Nodes ◽  
Lagrange Interpolation Polynomials ◽  
Interpolation Polynomials ◽  
Exact Rate Of Convergence

A well-known result due to S.N. Bernstein is that sequence of Lagrange interpolation polynomials for |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In this paper we present a quantitative version concerning the divergence behaviour of the Lagrange interpolants for |x|3 at equidistant nodes. Furthermore, we present the exact rate of convergence for the interpolatory parabolas at the point zero.

INFINITE PRODUCTS OF CYCLOTOMIC POLYNOMIALS

2015 ◽  
Vol 91 (3) ◽  
pp. 400-411 ◽  
Author(s):  
WILLIAM DUKE ◽  
HA NAM NGUYEN
Keyword(s):  
Asymptotic Behaviour ◽  
Dirichlet Series ◽  
Unit Circle ◽  
Roots Of Unity ◽  
Cyclotomic Polynomials ◽  
Natural Boundary ◽  
Infinite Products

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

scholarly journals Analytical and numerical aspect of coincidence point problem of quasi-contractive operators

2019 ◽  
Vol 105 (119) ◽  
pp. 101-121
Author(s):  
Faik Gürsoy ◽  
Müzeyyen Ertürk ◽  
Abdul Khan ◽  
Vatan Karakaya
Keyword(s):  
Convergence Rate ◽  
Strong Convergence ◽  
Rate Of Convergence ◽  
Metric Space ◽  
Iteration Method ◽  
Coincidence Point ◽  
Point Problem ◽  
Convex Metric Space ◽  
Numerical Examples ◽  
Strong Convergence Rate

We propose a new Jungck-S iteration method for a class of quasi-contractive operators on a convex metric space and study its strong convergence, rate of convergence and stability. We also provide conditions under which convergence of this method is equivalent to Jungck-Ishikawa iteration method. Some numerical examples are provided to validate the theoretical findings obtained herein. Our results are refinement and extension of the corresponding ones existing in the current literature.

Preparatory Lemmas

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Rate Of Convergence ◽  
Smooth Functions ◽  
Multi Index ◽  
The Moment

This chapter prepares for the proof by introducing a method concerning the general rate of convergence of mollifiers. The lemma takes into account the multi-index, the moment vanishing conditions, and smooth functions. An explanation for reducing the number of minus signs appearing in the proof is offered. The case N = 2 of the above lemma suffices for the proof of the main theorem. The chapter considers another way to work out the details relating to the lemma, which will be repeatedly used in the remainder of the proof. In particular, it describes functions whose integrals are not normalized to 1, but which satisfy the same type of estimates as ∈subscript Element.

scholarly journals On Lagrange interpolation at disturbed roots of unity

1993 ◽  
Vol 336 (2) ◽  
pp. 817-830 ◽  
Author(s):  
Charles K. Chui ◽  
Xie Chang Shen ◽  
Le Fan Zhong
Keyword(s):  
Lagrange Interpolation ◽  
Roots Of Unity

Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed A. Abdelkawy
Keyword(s):  
Brownian Motion ◽  
Error Analysis ◽  
Integral Equations ◽  
Lagrange Interpolation ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Numerical Examples ◽  
Collocation Technique

Abstract This paper addresses a spectral collocation technique to treat the stochastic Volterra–Fredholm integral equations (SVF-IEs). The shifted Legendre–Gauss–Radau collocation (SL-GR-C) method is developed for approximating the FSV-IDEs. The principal target in our technique is to transform the SVF-IEs to a system of algebraic equations. For computational purposes, the Brownian motion W(x) is discretized by Lagrange interpolation. While the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Some numerical examples are given to test the accuracy and applicability of our technique. Also, an error analysis is introduced for the proposed method.

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