scholarly journals Approximation by interpolation: the Chebyshev nodes

2021 ◽  
pp. 39-53
Author(s):  
Mama Foupouagnigni ◽  
Daniel Duviol Tcheutia ◽  
Wolfram Koepf ◽  
Kingsley Njem Forwa
Keyword(s):  
Chebyshev Nodes

scholarly journals On Hermite-Fejér type interpolation on the Chebyshev nodes

1993 ◽  
Vol 47 (1) ◽  
pp. 13-24 ◽  
Author(s):  
Graeme J. Byrne ◽  
T.M. Mills ◽  
Simon J. Smith
Keyword(s):  
Approximation Error ◽  
Interpolation Polynomial ◽  
Precise Estimate ◽  
Type Interpolation ◽  
Interpolation Methods ◽  
Chebyshev Nodes ◽  
Interpolatory Process

Given f ∈ C [−1, 1], let Hn, 3(f, x) denote the (0,1,2) Hermite-Fejér interpolation polynomial of f based on the Chebyshev nodes. In this paper we develop a precise estimate for the magnitude of the approximation error |Hn, 3(f, x) − f(x)|. Further, we demonstrate a method of combining the divergent Lagrange and (0,1,2) interpolation methods on the Chebyshev nodes to obtain a convergent rational interpolatory process.

scholarly journals Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous–Chebyshev nodes

2017 ◽  
Vol 43 ◽  
pp. 1-27 ◽  
Author(s):  
Peter Dencker ◽  
Wolfgang Erb ◽  
Yurii Kolomoitsev ◽  
Tetiana Lomako
Keyword(s):  
Polynomial Interpolation ◽  
Lebesgue Constants ◽  
Polyhedral Sets ◽  
Chebyshev Nodes

scholarly journals Optimal sampling rates for approximating analytic functions from pointwise samples

2018 ◽  
Vol 39 (3) ◽  
pp. 1360-1390 ◽  
Author(s):  
Ben Adcock ◽  
Rodrigo B Platte ◽  
Alexei Shadrin
Keyword(s):  
Least Squares ◽  
Analytic Functions ◽  
Exponential Convergence ◽  
Sampling Rate ◽  
Arbitrary Point ◽  
Impossibility Theorem ◽  
Weight Functions ◽  
Remez Algorithm ◽  
Chebyshev Nodes ◽  
Methods Numerical

AbstractWe consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best possible convergence rate of a stable method is root-exponential in M, and that any method with faster exponential convergence must also be exponentially ill conditioned at a certain rate. This result hinges on a classical theorem of Coppersmith & Rivlin concerning the maximal behavior of polynomials bounded on an equispaced grid. In this paper, we first generalize this theorem to arbitrary point distributions. We then present an extension of the impossibility theorem valid for general nonequispaced points and apply it to the case of points that are equidistributed with respect to (modified) Jacobi weight functions. This leads to a necessary sampling rate for stable approximation from such points. We prove that this rate is also sufficient, and therefore exactly quantify (up to constants) the precise sampling rate for approximating analytic functions from such node distributions with stable methods. Numerical results—based on computing the maximal polynomial via a variant of the classical Remez algorithm—confirm our main theorems. Finally, we discuss the implications of our results for polynomial least-squares approximations. In particular, we theoretically confirm the well-known heuristic that stable least-squares approximation using polynomials of degree N < M is possible only once M is sufficiently large for there to be a subset of N of the nodes that mimic the behavior of the $N$th set of Chebyshev nodes.

scholarly journals Solution of inverse heat conduction equation with the use of Chebyshev polynomials

2016 ◽  
Vol 37 (4) ◽  
pp. 73-88 ◽  
Author(s):  
Magda Joachimiak ◽  
Andrzej Frąckowiak ◽  
Michał Ciałkowski
Keyword(s):  
Inverse Problem ◽  
Chebyshev Polynomials ◽  
Direct Problem ◽  
Least Square Method ◽  
Least Square ◽  
Random Disturbance ◽  
Inverse Heat Conduction ◽  
Chebyshev Nodes ◽  
The Difference ◽  
The Stability

AbstractA direct problem and an inverse problem for the Laplace’s equation was solved in this paper. Solution to the direct problem in a rectangle was sought in a form of finite linear combinations of Chebyshev polynomials. Calculations were made for a grid consisting of Chebyshev nodes, what allows us to use orthogonal properties of Chebyshev polynomials. Temperature distributions on the boundary for the inverse problem were determined using minimization of the functional being the measure of the difference between the measured and calculated values of temperature (boundary inverse problem). For the quasi-Cauchy problem, the distance between set values of temperature and heat flux on the boundary was minimized using the least square method. Influence of the value of random disturbance to the temperature measurement, of measurement points (distance from the boundary, where the temperature is not known) arrangement as well as of the thermocouple installation error on the stability of the inverse problem was analyzed.

Fast factorization of rectangular Vandermonde matrices with Chebyshev nodes

2018 ◽  
Vol 81 (2) ◽  
pp. 547-559 ◽  
Author(s):  
Mykhailo Kuian ◽  
Lothar Reichel ◽  
Sergij V. Shiyanovskii
Keyword(s):  
Chebyshev Nodes ◽  
Vandermonde Matrices

scholarly journals Hermite-Fejer interpolation at the ‘practical’ Chebyshev nodes

1973 ◽  
Vol 9 (3) ◽  
pp. 379-390
Author(s):  
R.D. Riess
Keyword(s):  
Continuous Functions ◽  
Negative Results ◽  
Chebyshev Nodes ◽  
Chebyshev Points ◽  
Uniformly Convergent ◽  
Lagrangian Interpolation

Berman has raised the question in his work of whether Hermite-Fejér interpolation based on the so-called “practical” Chebyshev points, , 0(1)n, is uniformly convergent for all continuous functions on the interval [−1, 1]. In spite of similar negative results by Berman and Szegö, this paper shows this result is true, which is in accord with the great similarities of Lagrangian interpolation based on these points versus the points , 1(1)n.

scholarly journals On Berman's phenomenon in interpolation theory

1975 ◽  
Vol 12 (3) ◽  
pp. 457-465 ◽  
Author(s):  
W. Lyle Cook ◽  
T.M. Mills
Keyword(s):  
Higher Order ◽  
Interpolation Process ◽  
Divergence Theorem ◽  
Interpolation Theory ◽  
Open Problems ◽  
Chebyshev Nodes

In 1965, D.L. Berman established an interesting divergence theorem concerning Hermite-Fejér interpolation on the extended Chebyshev nodes. In this paper it is shown that this phenomenon is not an isolated incident. A similar divergence theorem is proved for a higher order interpolation process. The paper closes with a list of several related open problems.

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