# 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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### Is there something like “associated” Chebyshev polynomials?

When I was experimenting with orthogonalization of polynomials $$p_n(x)=\begin{cases} 1-x^n&\text{if }n\equiv0\; (\operatorname{mod}2),\\ x-x^n&\text{otherwise}, \end{cases}$$ i.e. simplest ...
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### Estimating Spline curve by OLS. Is a good idea to fix the knots at Chebyshev sites?

I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or ...
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### Probabilties using Central Limit Theorem

Let n be an independent random variables and the number of orders in a 120 minute period. Given that $\mu$ is 1.5 minutes and that $\sigma$ is 1 minute use the Central Limit Theorem to find the ...
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### Complex Chebyshev Polynomials

Chebyshev Polynomials can be used to compute a very nearly minimax polynomial approximation of an analytic function on $[-1,1]$. Is there a complex analog that can compute a nearly minimax polynomial ...
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### Numerical analysis: Chebyshev coefficient representation error.

I am unsure if numerical analysis questions are suitable for this forum, but I have nowhere else to ask, so if this question is inappropriate, tell me and I will delete it. If $x_k$ are the Chebyshev ...
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### Pythagorean-like equation for generalized hyperbolic function

Trig functions satisfy $\cos^2t+\sin^2t=1$, which is an expression of the Pythagorean theorem. Hyperbolic trig functions satisfy $\cosh^2t-\sinh^2t=1$ which may perhaps be viewed as a generalization. ...
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### Demystifying math: how could someone come up with Chebychev polynomials?

I hope this question is allowed, I am interested how you think someone could come up with the Chebychev polynomials, where I refer to them in the sense that someone would be interested in the ...
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### Generating Chebyshev polynomials by Gram-Schmidt

Given the definition of Chebyshev polynomials in this form: $$T_n(x) = \cos(n\cos^{-1}x), n\ge 1, T_0=1$$ I want to show that using Gram-Schmidt procedure with set $\{1, x, x^2, \dots\}$ and weight ...
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### Constraints on a Chebyshev series representation of a CDF

My question is about deriving constraints for coefficients of a Chebyshev series which represents a CDF. Let $F(x)$ be the cumulative distribution function for $x\in [-1,1]$. Accordingly we know ...
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### Use 1-degree Chebyshev polynom to approximate $\cos(x)$ and calculate the error

The task is to give for $\cos(x)$ the nodes of the interpolation polynom of degree 1 that approximates the function on $[-\pi,\pi]$ the best as well as the related error. I want to solve this task ...
The Chebyshev polynomials of the first kind $T_n(x)$ are known to form a complete orthogonal basis for functions on $[-1,1]$. I was looking for a proof of the completeness part, without any luck, when ...
Suppose that $f \in C[-1,1]$ has a chebyshev series $\sum_{n=1}^{\infty}a_nT_n$ (b) show that $E_n(T_{n+1})=1$ (c) show that $|E_n(f)-|a_{n+1}|| \le \sum_{k=n+2}^{\infty}|a_k|$ cf : \$E_n(f)= \inf\{||...