# 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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### A question about extending polynomial span to monomial basis

I have a final next week and our instructor gave us some examples with solutions but I could not understand some operations. Inner product is $$(p,q)=\int_{-1}^{1} p(t)q(t)dt$$ $W = Span\{1,t,t^2\}$ ...
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### Is this true for $n\geq 0$: $\lim\limits_{x\to \infty} e^{-x^2}\frac d {dx}\frac{T_n}{T_{n+1}}e^{x^2}=1$?

Let $T_n$ denote Chebyshev polynomials of the first kind, I w'd like to show the below limit if it is true which it's hold for few first of them then how do i show this if it is true for $n\geq0$ ...
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### Finding Points for Polynomial Interpolation

I'm studying for a final and came across the following question: Question In a discussion with my professor, he said using Chebyshev polynomials would be messy and unwieldy and encouraged another ...
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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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### Chebyshev polynomials increase more quickly than any other polynomial outside $[-1,1]$

In Appendix C3 of Shewchuk's excellent notes on conjugate gradient, it is stated without proof that Chebyshev polynomials... increase in magnitude more quickly outside the range $[-1,1]$ than any ...
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### Any differences between Lagrange polynomial on Chebyshev points and Chebyshev polynomial?

I have to find an approximate continuous function that passes through a number of points. Many have said that the best (for my specific problem) is to use Chebyshev polynomial decomposition. I have ...
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### Are there any alterations for the Chebyshev Differentiation Matrices on an arbitrary domain [a,b]?

I'm implementing the Chebyshev collocation method for solving PDEs, more specifically the shallow water equations. I know the Chebyshev differentiation matrices (or differential operators) are, for ...
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### Finding the dual of a LP

I am trying to find the dual of the following linear program. The primal LP's purpose is to find the lowest possible L1 (sum of absolute values of coefficients) of a degree $d$ polynomial such that (1)...
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### Reverse economization of Chebyshev series

Suppose I have some function which is represented as converging series of Chebyshev polynomials of first kind in $[-1;1]$: $$f(x)=\sum\limits_{n=1}^\infty a_n T_{2n}(x)$$ I need to transform this ...
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### Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
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### Cubic Approximation to $e^x$ using Chebyshev Polynomial

Was trying to solve this: $C_r=\frac{2}{\pi}\int_{-1}^1\frac{e^xT_r(x)}{\sqrt{1-x^2}}dx$ where $r=0,1,2,3$ $T_r(x) =cosr[{cos}^{-1}x]$ While solving, I equated $x=cos\theta$ Therefore ...
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### Inverse Chebyshev Recurrence

The Chebyshev polynomials (of the first kind) are a sequence of polynomials defined recursively by $$\begin{cases} T_{0}(x) = 1 \\ T_{1}(x) = x \\ T_{n}(x) = 2xT_{n-1}(x) - T_{n-2}(x) \end{cases}$$ ...
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### Is this polynomial irreducible over the rationals?

Prove (or disprove): Define $T_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$. If $p$ is an odd prime, then $\sqrt{\frac{T_p(x)-1}{x-1}}$ is an irreducible polynomial over the ...
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### Transform complex trigonometric expression with $\arccos$

In the proof that the poles of a Chebyshev filter lie on an ellipse, there is the following transformation, for the $s$ values correspondant to the poles. From (1) s_{pm} = j \cos \left[ \...
I'm looking into SLATEC implementation of Bessel function $J_0$ computation (readable in C in GSL), namely at its part for arguments in interval $(0,4)$. There a Chebyshev expansion is used, but the ...
Carl de Boor poses the following problem in his A Practical Guide to Splines (1978 - Chapter II, p. 38, problem 4): The calculation of $||g|| = \max\{|g(x)| : a \le x \le b\}$ is a nontrivial task. ...