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They are used as an approximation to a least squares fit, and are a special case of the Gegenbauer polynomial with alpha=0. They are also intimately ...

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined several equivalent ways; in this article the polynomials are defined by starting with trigonometric functions: The Chebyshev polynomials of the first kind are given by Similarly, define the Chebyshev polynomials of the second kind as

approximation of an arbitrary function of x in the interval [-1,1] by calculating the coefficients cj at the N zero's xk of the N-th Chebyshev polynomial:.

approximations. Moreover Chebyshev polynomials may be used as a method of minimization of map projection distortion. The example of such projection is shown in the paper. 1. Uniform approximation . Approximation performed with the use of Chebyshev polynomials is called „the uniform approximation”.

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

where the Chebyshev polynomials Ti (of the 1st kind) are described by the recursion: Tn+1(x) ...

In this chapter we describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to compute efficiently such ...

1. Uniform approximation . Approximation performed with the use of Chebyshev polynomials is called „the uniform approximation”. It consists of approximation of the function f(x) by the polynomial n(xT) in the interval . ∈, x a b, in such a way, that the deviation of the highest absolute value ( ) n (E f x T x) x a b = −. ∈ , max (1)

The Chebyshev approximation (5.8.9) is very nearly the same polynomial as that holy grail of approximating polynomials the minimax polynomial, which (among all polynomials of the same degree) has the smallest maximum deviation from the true function f(x). The minimax polynomial is very difficult to find; the Chebyshev

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

In the Chebyshev approximation, the average error can be large but the maximum error is minimized. Chebyshev approximations of a function are sometimes said to ...

To approximate a function by a linear combination of the first N Chebyshev polynomials (k=0 to N-1), the coefficient ck c k is simply equal to A ...

CHEBYSHEV POLYNOMIAL APPROXIMATION TO SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS By Amber Sumner Robertson May 2013 In this thesis, we develop a method for nding approximate particular so-lutions for second order ordinary di erential equations. We use Chebyshev polynomials to approximate the source function and the particular solution of

Chebyshev polynomials are important in approximation theory because the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points ...

best approximating polynomial. In chapter 5 we will explain what Cheby-shev polynomials are, since we need them to nd the best approximating polynomial in chapter 6. In chapter 6 we show Chebyshev’s solution to the approximation problem, compare this to …

We use two different basis functions (i.e. Chebyshev basis and spline basis). The approximation error is shown in Figures (1-2). Insert Figures (1-2) here As can be seen, when the approximation is calculated using Chebyshev basis functions the error is of the order of 1×10-15 for a polynomial of order 20.

CHEBYSHEV POLYNOMIAL APPROXIMATION TO SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS By Amber Sumner Robertson May 2013 In this thesis, we develop a method for nding approximate particular so-lutions for second order ordinary di erential equations. We use Chebyshev polynomials to approximate the source function and the particular solution of

Chebyshev polynomials are important in approximation theory because the roots of T n (x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation.

Particular emphasis is given to Chebyshev polynomials, with brief applications to electric circuit theory. 1. The Problem of Approximation. Chebyshev Approximation. Consider a function /(x) defined in an interval a ^ x S b.

We use Chebyshev polynomials to approximate the source function and the particular solution of an ordinary differential equation. The derivatives of each ...