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 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 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.

scholarly journals A note on optimal Hermite interpolation in Sobolev spaces

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Guiqiao Xu ◽  
Xiaochen Yu
Keyword(s):  
Weight Function ◽  
Sobolev Spaces ◽  
Lagrange Interpolation ◽  
Hermite Interpolation ◽  
Weighted Spaces ◽  
Interpolation Algorithms ◽  
Interpolation Systems

AbstractThis paper investigates the optimal Hermite interpolation of Sobolev spaces $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] , $n\in \mathbb{N}$ n ∈ N in space $L_{\infty }[a,b]$ L ∞ [ a , b ] and weighted spaces $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p< \infty $ 1 ≤ p < ∞ with ω a continuous-integrable weight function in $(a,b)$ ( a , b ) when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in $L_{\infty }$ L ∞ (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ) are optimal for $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] in $L_{\infty }[a,b]$ L ∞ [ a , b ] (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.

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.

Interpolation Method Research and Precision Analysis of GPS Satellite Position

2018 ◽  
Vol 6 (3) ◽  
pp. 277-288
Author(s):  
Jianmin Wang ◽  
Yabo Li ◽  
Huizhong Zhu ◽  
Tianming Ma
Keyword(s):  
Lagrange Interpolation ◽  
Hermite Interpolation ◽  
Interpolation Method ◽  
Spline Interpolation ◽  
Cubic Spline ◽  
Newton Interpolation ◽  
Cubic Spline Interpolation ◽  
Satellite Position ◽  
Precise Ephemeris ◽  
Better Than

Abstract According to the precise ephemeris has only provided satellite position that is discrete not any time, so propose that make use of interpolation method to calculate satellite position at any time. The essay take advantage of IGS precise ephemeris data to calculate satellite position at some time by using Lagrange interpolation, Newton interpolation, Hermite interpolation, Cubic spline interpolation method, Chebyshev fitting method respectively, which has a deeply analysis in the precision of five interpolations. The results show that the precision of Cubic spline interpolation method is the worst, the precision of Chebyshev fitting is better than Hermite interpolation method. Lagrange interpolation and Newton interpolation are better than other methods in precision. Newton interpolation method has the advantages of high speed and high precision. Therefore, Newton interpolation method has a certain scientific significance and practical value to get the position of the satellite quickly and accurately.

scholarly journals Fractional Hermite interpolation for non-smooth functions

2020 ◽  
Vol 52 ◽  
pp. 113-131
Author(s):  
Jiayin Zhai ◽  
Zhiyue Zhang ◽  
Tongke Wang
Keyword(s):  
Hermite Interpolation ◽  
Smooth Functions

scholarly journals Matrix Transformations and Disk of Convergence in Interpolation Processes

2008 ◽  
Vol 2008 ◽  
pp. 1-11
Author(s):  
Chikkanna R. Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Lagrange Interpolation ◽  
Hermite Interpolation ◽  
Matrix Transformation ◽  
Matrix Transformations ◽  
Taylor Polynomial ◽  
Lagrange Polynomial ◽  
Birkhoff Interpolation ◽  
Partial Sum ◽  
The Difference ◽  
Definition Of

Let denote the set of functions analytic in but not on . Walsh proved that the difference of the Lagrange polynomial interpolant of and the partial sum of the Taylor polynomial of converges to zero on a larger set than the domain of definition of . In 1980, Cavaretta et al. have studied the extension of Lagrange interpolation, Hermite interpolation, and Hermite-Birkhoff interpolation processes in a similar manner. In this paper, we apply a certain matrix transformation on the sequences of operators given in the above-mentioned interpolation processes to prove the convergence in larger disks.

scholarly journals Local Lagrange Interpolations Using BivariateC2Splines of Degree Seven on Triangulated Quadrangulations

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Xuqiong Luo ◽  
Qikui Du
Keyword(s):  
Lagrange Interpolation ◽  
Order Approximation ◽  
Optimal Order ◽  
Interpolation Scheme ◽  
Smooth Functions

A local Lagrange interpolation scheme using bivariateC2splines of degree seven over a checkerboard triangulated quadrangulation is constructed. The method provides optimal order approximation of smooth functions.

FRACTAL BASES FOR BANACH SPACES OF SMOOTH FUNCTIONS

2015 ◽  
Vol 92 (3) ◽  
pp. 405-419 ◽  
Author(s):  
M. A. NAVASCUÉS ◽  
P. VISWANATHAN ◽  
A. K. B. CHAND ◽  
M. V. SEBASTIÁN ◽  
S. K. KATIYAR
Keyword(s):  
Hermite Interpolation ◽  
Iterated Function Systems ◽  
Smooth Functions ◽  
Bounded Linear ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Scaling Parameters ◽  
Fractal Functions ◽  
Present Setting ◽  
Interpolation Functions

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called $\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space ${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for ${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.

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