# 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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### Roots of the Chebyshev polynomials of the second kind.

It is known that the roots of the Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$ and they are simple (of multiplicity one). I have noticed that the roots ...
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### Prove that $\int_1^a \frac{T_n(x) T_n(x/a)}{\sqrt{a^2 - x^2} \sqrt{x^2 - 1^2}} \frac{a}{x} \mathrm{d}x = \frac{\pi}{2}$

In the paper, Representation of a Function by Its Line Integrals, with Some Radiological Applications, A. M. Cormack, Journal of Applied Physics 34, 2722 (1963), an integral identity is expressed ...
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### How do you define the sample points used for Chebyshev approximation/interpolation?

It appears there are somewhat conflicting definitions of the points used in Chebyshev interpolation. Wikipedia and Numerical Recipes define the $x_j^{(n)}$ sample points for $(n-1)^\text{th}$-order ...
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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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### Integrating Chebyshev polynomial of the first kind

I'm trying to evaluate the integral of the Chebyshev polynomials of the first kind on the interval $-1 \leq x \leq 1$ . My idea is to use the closed form $$T_n(x) = \frac{z_1^n + z_2^{-n} }{2}$$ ...
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### Extrema of Chebyshev polynomials (of the first kind)

I can hardly find a proof why the extrema of the Chebyshev polynomials are $$x_k=\cos(\frac{k}{n}\pi), k=1,...n$$ and also why there are $n+1$ of them. The Chebyshev polynomials are here defined as ...
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### show that nth Chebyshev polynomial is an nth order polynomial

Define the Chebyshev polynomial $T_n(x)=\cos(n\cos^{-1}(x)), n\geq 1, T_0=1)$. Show that $T_n(x)$ is an nth order polynomial This is my attempt, however I couldn't reduce it to a polynomial. \...
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### Limit of Ratio of Chebyshev Polynomials

I have been trying to compute the limit $$\lim_{n\to\infty}{{U_n(x)^2}\over{U_{n-1}(x)^2+U_n(x)^2}}$$ where $U_n(x)$ is the $n$-th Chebyshev polynomial of the second kind and $x\ge 1$. Using software ...
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### What is a tensor-product Chebyshev grid?

What is the difference between "Chebyshev grid" and "tensor-product Chebyshev grid"? Are they defined on a 2D vector?
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### The rate of convergence for polynomial interpolation of different functions

I have written some code in Matlab to polynomially interpolate functions with Chebyshev nodes using the Chebyshev base ( I calculate the coefficients with respect to the Chebyshev base and multiply ...
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I recently had the need to approximate this function $$f\left(x\right)=\begin{cases} \log\left(\frac{\pi+2\arcsin\left(x\right)}{\pi}\right), & x<0\\ -\log\left(2-\frac{\pi+2\arcsin\left(x\... 0answers 150 views ### From polynomials to Chebyshev polynomials I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here: In order to obtain “strong” stability, we replace the condition (2.7) by$$\left|\prod_{j=1}^N (1-\...
I would like to know the proof of $$\int_a^b \frac{T_n(x/a)T_n(x/b)\, dx}{x(b^2-x^2)^{1/2}(x^2-a^2)^{1/2}}=\frac{\pi}{2 ab}, 0<a<b, n \in \Bbb N$$ where $T_n(x)$ is the Chebyshev polynomial of ...