Questions tagged [chebyshev-polynomials]

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

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Uniqueness of minimal $\infty$-norm polynomial

From this proof it is clear to me that Chebyshev polynomial $\frac{1}{2^{n-1}} T_n(x)$ is minimum $\infty$-norm in $[-1,1]$ among the monic polynomials of degree $n$. How to prove the uniqueness (if ...
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Bound 1-norm of Chebyshev coefficients in terms of supremum norm of function

Is there a constant $C$ such that $\|c_k\|_{1} \leq C \, \|f\|_\infty$ with $c_k$ the Chebyshev coefficients of $f$? I'm assuming the answer is no, but I can't find a counterexample.
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Chebyshev Polynomials: Properties of Derivatives

Show that: $T_n'(x)$=$2n\sum_{k=0\\k+n~~odd}^{n-1}\frac{1}{c_k}T_k(x)$ $T_n''(x)$=$\sum_{k=0\\k+n even}^{n-2}\frac{1}{c_k}n(n^2-k^2)T_k(x)$ where $c_0=2$ and $c_n=1$ for $n\geq1$ I tried using the ...
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Why use chebyshev polynomial in this problem?

$f(x)$ is polynomial degree of 6. For $-1=<x=<1$ , $0=<f(x)=<1$ . What is maximum value of leading coefficient of $f(x)$. I saw solution, solution claim $g(x)=2f(x) -1$ and $g(x)=T_6 (x)$...
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Interpolation on Chebyshev point with octave

I have to solve this numeric problem on octave: (A) Check the correctness of the Lagrange (or Newton) interpolation method on some functions, of which the analytical formula is known, considering ...