On the asymptotic constant for the rate of Hermite–Fejér convergence on Chebyshev nodes

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.

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An upper bound for the rate of convergence of the Hermite-Fejér process on the extended Chebyshev nodes of the second kind

Journal of Approximation Theory ◽

10.1016/0021-9045(79)90057-1 ◽

1979 ◽

Vol 26 (3) ◽

pp. 195-203 ◽

Cited By ~ 7

Author(s):

R Bojanic ◽

J Prasad ◽

R.B Saxena

Keyword(s):

Rate Of Convergence ◽

Upper Bound ◽

Chebyshev Nodes

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

Journal of Complexity ◽

10.1016/j.jco.2017.05.001 ◽

2017 ◽

Vol 43 ◽

pp. 1-27 ◽

Cited By ~ 3

Author(s):

Peter Dencker ◽

Wolfgang Erb ◽

Yurii Kolomoitsev ◽

Tetiana Lomako

Keyword(s):

Polynomial Interpolation ◽

Lebesgue Constants ◽

Polyhedral Sets ◽

Chebyshev Nodes

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Discretization of Dynamic Loading in Time History Analysis

Journal of Mechanics ◽

10.1017/jmech.2013.68 ◽

2013 ◽

Vol 30 (1) ◽

pp. 57-65

Author(s):

S.-Y. Chang

Keyword(s):

Dynamic Loading ◽

Amplification Factor ◽

Integration Method ◽

Time History ◽

Discretization Error ◽

Amplitude Error ◽

History Analysis ◽

Asymptotic Constant ◽

Error Amplification ◽

General Step

ABSTRACTAlthough the numerical properties of a step-by-step integration method can be evaluated based on the currently available techniques, there is still lack of a technique for evaluating its capability to capture dynamic loading. In this work, the amplitude error caused by the step discretization error is identified and the correlation between the relative amplitude error and relative step discretization error is analytically established. As a result, it is thoroughly confirmed that the asymptotic constant of the discretization error amplification factor for the displacement response to a cosine loading can be considered as an indicator of the capability to capture dynamic loading for a general step-by-step integration method.

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Optimal sampling rates for approximating analytic functions from pointwise samples

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry024 ◽

2018 ◽

Vol 39 (3) ◽

pp. 1360-1390 ◽

Cited By ~ 4

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.

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

Archives of Thermodynamics ◽

10.1515/aoter-2016-0028 ◽

2016 ◽

Vol 37 (4) ◽

pp. 73-88 ◽

Cited By ~ 4

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.

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