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point of "no-error" or "zero error" to deal with , needs to carry on the promotion to Chebyshev optimal approximation principle. For promotion of Chebyshev polynomials, [1] gave two . Manuscript received April 12, 2013; revised June 29, 2013. Lemin Gu is with the Tongji University, China (e-mail: gulemin@tongji.edu.cn).

Error between optimal polynomial and exp(x) (red), and Chebyshev approximation and exp(x) (blue) over the interval [−1, 1].

It is well known that a near minimax polynomial approximation p is obtained by truncating the Chebyshev series of a function ƒ; after n + 1 terms.

Rational Chebyshev Approximations ... Clenshaw's [3] Chebyshev series ex- ... error can be avoided by evaluating erf (.r) directly and erfc (x) indirectly ( ...

2.2 Chebyshev’s interest in approximation theory Chebyshev was since his childhood interested in mechanisms. The theory of mechanisms played in that time an important role, because of the industri-alisation. In 1852, he went to Belgium, France, England and Germany to talk with mathematicians about di erent subjects, but most important for him ...

uniform norm give the same best approximating polynomial. Keywords: approximation theory, Chebyshev, L2-norm, uniform norm, algebraic polynomial, error ...

I am trying to approximate a function f(x) on [−1,1] using Chebyshev's polynomial of the first kind. f(x)≈N∑i=0aiTi(x). What is the error ...

|| f – p ||∞. The Mathematica function MiniMaxApproximation minimizes the relative error by minimizing. || (f – p) / f ||∞. As was ...

Chebyshev Approximations¶. This chapter describes routines for computing Chebyshev approximations to univariate functions. A Chebyshev approximation is a truncation of the series , where the Chebyshev polynomials provide an orthogonal basis of polynomials on the interval with the weight function .The first few Chebyshev polynomials are, ,

5.8 Chebyshev Approximation The Chebyshev polynomial of degree n is denoted Tn(x), and is given by the explicit formula Tn(x)=cos(n arccos x)(5.8.1) This may look trigonometric at first glance (and there is in fact a close relation between the Chebyshev polynomials and the discrete Fourier transform); however

3 Errors in Truncating Infinite Series. 11. 4 Equivalence Theorem. 18. 5 Interpolation & Aliasing Errors in Chebyshev Polynomial Co-.

using the roots of the Chebyshev polynomials Numerical Analysis (MCS 471) Chebyshev Points & Padé Approximations L-16 29 September 2021 4 / 35 the interpolation error

In this paper, we present new error estimates for polynomial interpolation in the Chebyshev points of the first and second kind, which is a ...

However, in attempting to find a suitable polynomial approximation to a general function f(x), the integral occurring in equation (1.3) cannot be evaluated ...

in the Chebyshev points of the flrst or second kind does not sufier from the Runge phenomenon ([19], pp. 146), which makes it much better than the interpolant in equally spaced points, and the accuracy of the approximation can improve remarkably fast when the number of interpolation