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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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$\cos\frac\pi{n}$ Analytic expression

I recently found out that $$\sin\frac\pi5=\frac12\sqrt{\frac{5-\sqrt5}2}$$ Which means that $$\cos\frac\pi5=\frac{1+\sqrt5}4$$ I also recently found that if $n\in\Bbb N$, $$\sin nx=\sin x\,U_{n-1}(\...
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Finding $\int_{0}^{1} \frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{\sqrt{x-x^2}} dx$

Shifted Chebyshev polynomials $$T_{i}^{*}(x) = \cos(i \arccos(2x-1))$$ We want to calculate $$I=\int_{0}^{1} \frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{\sqrt{x-x^2}} dx$$ Which is equal to $$\sum_{\substack{...
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Is the following matrix defined by the roots of Chebyshev polynomial invertible?

Let $x_0, \dots , x_n$ the roots of the Chebyshev polynomial, $T_{n+1}(x)$. We define: $\begin{pmatrix} \frac{1}{\sqrt2}T_0(x_0) & \cdots & \frac{1}{\sqrt2}T_0(x_0) \\ T_1(x_0) & \...
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How has this Chebyshev expansion been reindexed?

Just a quick question, I'm going through my lecture notes and I can't see how the author has gone from this: $$\begin{aligned} f ( x ) g ( x ) & = \sum _ { m = 0 } ^ { \infty } \breve { f } _ { m }...
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Chebychev's Inequality question… need help?

Please could I have some help with the following question? My initial way of thinking was that Ui must be less than $5$ so that the measurement of the melting point is within $5 $ degrees of $c$, so I ...
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Chebyshev polynomials and trace of $A \in SL_2(\mathbb{C})$

Defining $C_n(z) = \frac{z^m + z^{-m}}{2}$, the Chebyshev polynomials are defined by $$T_n(C_1(z)) = C_n(z)$$ and are given by $T_1(z) = z, T_2(z) = 2z^2-1, T_3(z) = 4z^3-3z$, etc. Since for $...
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Chebyshev coefficients- interpolation on [a,b]

My problem is to solve a second order differential equation given two (Dirichlet) boundary conditions. $\frac{d^2y}{dx^2} = M/EI$ Both M and I are functions of x. Owing to complexity of the ...
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Chebyshev approximation for large interval

In the context of neural networks and cryptography, I would like to approximate some activation functions. However, I need to approximate them into polynomial forms for my purpose. It seems that ...
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Solving an integral with four point Gauss-Chebyshev

I am struggling with this question as in our notes it shows how to solve Gauss-Chebyshev integrals of the form $ \int_{-1}^{1}\frac{f(x)}{\sqrt{1-x^2}}dx$ , however this is different. $$\text{Solve ...
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Proving that $T_0, T_1, T_2, …$ are basis of $\mathbb{R}[x]$

I am given that the Chebyshev polynomial $T_n(x) \in \mathbb{Q}[x]$ is a polynomial such that $T_0(x) = 1$, $T_1(x) = x$ and for $n \ge 2$, $T_n(x) = 2xT_{n-1}(x)-T_{n-2}(x)$ Now, I am supposed to ...
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What is this generalization of the Chebyshev polynomials?

For $\varepsilon>0$ consider the tridiagonal matrix $$L_{\varepsilon}=\begin{bmatrix} 0 & 1 & \ & \ & \ & \ & \ & \ \\ 1 & \varepsilon & 1 & \ & \ &...
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Maximum values of Chebyshev Polynomials

Let $T_n(x)$ be the $nth$ Chebyshev Polynomial. Prove that $|T_n(x)| \leq 1$ for $x \in [0,1]$ and determine when there is equality. So far I was thinking that $t$ would have to be of the form $\frac{...
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Minimal error chebyshev interpolation

Let's say the n-degree Chebyshev polynomials : $$ T_{n} (x)=\cos(n\arccos(x))$$ Make a polynomial such that: $$\mid y- P (x) \mid$$ be minimal, using the first three Chebyshev polynomials for the ...
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This has to be in the lit somewhere. Can someone point me to this in any accessible book or lit?

It's just a big trig, sinusoidal, Fourier series thing: $$\begin{align} y(t) &= \sum_{k=0}^{K} a_k \big( A \cos(\omega t) \big)^k \\ \\ &= \sum_{n=0}^{K} b_n \cos(n \omega t) \\ \end{...
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Linear vs Orthogonal polynomial for fitting

I am neither a mathematician nor do I possess a large knowledge in maths so this could also be completely false. But I read that if you use an orthogonal function like the Chebychev-polynomial to ...
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What is the group-like structure on $x^2+y^2+z^2-2xyz=1$?

(Background: this is inspired by Chebyshev polynomials and expanding a function as a Chebyshev series.) Solving for $ z $ gives $$ z=xy \pm \sqrt{(1-x^2)(1-y^2)}, $$ where $-1\leq x,y \leq 1$. Now ...
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Showing That Chebyshev Polynomials Are Orthogonal

This is a problem in an upcoming lecture: Show that the first two Chebyshev polynomials, $T_0(x) = 1$ and $T_1(x) = x$ are orthogonal with respect to the weighting function $r(x) = (1 − x^2)^{-\...
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Chebyshev's polynomial approximation

Please help me with this question. I want to run a system that will give me answer using MATLAB but I am struggling. Write a function Cheb(n,x) for evaluating $T_n(x)$. Use the recursive formula ...
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Chebyshev polynomial: recursive formula error estimate

I am trying to solve Problem 3 & 4 from Numerical methods of Bakhvalov, Zhidkov, Kobelkov from section 2.8 on Chebyshev polynomials. If in the recursive formula $$ T_n(x) = 2xT_{n-1}(x)-T_{n-2}(x)...
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Limit superior of a sequence of oscillating functions related to Chebyshev polynomials

Let $n \in \mathbb N$ and consider the polynomial function $f_n \colon \mathbb R \to \mathbb R$ defined by $$f_n(x) = \sum_{k=0}^n (-1)^k \binom {2n+1} {2k+1} (1 - x^2)^{n-k} x^{2k}$$ for any $x \in \...
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$\Theta =70$ degrees into seven angles $\Theta /7=10$ is not possible by euclids method

Why we can not divide $\Theta =70$ degrees into seven angles $\Theta /7=10$ degrees using euclids tools. How I apply chebyshev polynomial to solve it. Does any one can help, I am not sure how to ...
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proof of chebychev polynomial

Can someone explain the Intuition or the reasoning of the following Chebychev Theorem (page 72 Theorem 2.7.2): In order that the ordinary polynomial $P(x)$ among all polynomials of degree < $n$ ...
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Chebyshev polynomial. Show : $ T_n(x) = \frac{1}{2} ((x+\sqrt{x^2-1})^n+(x-\sqrt{x^2-1})^n)$

Consider $T_n(x) = \cos ( n \cdot \arccos(x)) $ on $ I = [-1,1]$. Show: a: $T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x) $ b : The $T_n$ are orthogonal with $(f,g) = \int_{-1}^{1} f(x)g(x)\frac{1}{\sqrt{1-x^...
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Lagrangian interpolation at Chebyshev points - estimate on coefficients in monomic basis

First, let us fix some Notation: Let $n\in\mathbb{N}$ and $x_i=\cos(\tfrac{(i+1/2)\pi}{(n+1)})$, $i=0,\dots,n$, be the Chebyshev points. Let \begin{align}L_i(x)={\displaystyle\prod_{\substack{0\leq j\...
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Chebyshev coefficients of $e^{-x}$

Im trying to derive, given $n$, the Chebyshev coefficients $c_k$ of $e^{-x}$ on [-1,1]. That is $c_k=\frac{(e^{-x},T_k)_w}{(T_k,T_k)_w}$, $k=0,1,...,n.$ I have problems computing $(e^{-x},T_k)_w= \...
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Is there a general form of Chebyshev expansion coefficients for Gaussian distribution

$\newcommand{\chebyshevt}{\text{chebyshevt}}$ $\newcommand{\Norm}{\text{Norm}}$ I tried to calculate the coefficient for distribution $Norm(x, \mu, \sigma)$ via $$\int_{-1}^{1} \chebyshevt(x, t) \...
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Palindromic combinations of Chebyshev Polynomials share common roots?

Suppose that the real polynomial below $$p(x)=\sum_{k=0}^{2n}\alpha_{k}x^k$$ is a palindromic polynomial of even degree; that is, $p_{2n-k}=p_k$ for $0\leq k\leq 2n$ and $\alpha_0\neq 0$. Is it ...
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Finding error of a Gaussian-Chebyshev quadrature rule

Suppose we want to integrate $$I(f) := \int_{-1}^1{f(x)\over\sqrt{1-x^2}}\,dx.$$ If I have some quadrature formula given by $$Q_2(f) = {\pi\over 2}\sum_{i=1}^2f(x_i),$$ I want to put an upper bound on ...
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The complement of a tree in the complex plane is conformally equivalent to the complement of the closed unit disk .

Recently I'm reading about the relation between Shabat polynomials and tress . The book says that the pre image of a line segment in the complex plane under a polynomial is the union of n ...
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Show that a matrix is nonsingular [closed]

What is an efficient way of showing that the matrix $$\begin{align} P\triangleq \begin{bmatrix}\cos\theta_1&\sin\theta_1&...&\cos\theta_n&\sin\theta_n\\ \cos2\theta_1&\sin2\...
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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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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 ...
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On the extrema of Chebyshev polynomials of the second kind

I wish to prove that the magnitude of extreme values of $U_n(x)$, the Chebyshev polynomial of the second kind, is monotonically increasing on $[-1,1]$. By symmetry it suffices to prove it over $[0,1]$....
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Moments of a random variable in the Chebyshev basis

let $x\in [-1,1]$ be a random variable. We are given a finite sequence of moments of $x$, e.g., $E[x^{\alpha}], \alpha=0,...,N$. I want to find the moments in the Chebyshev basis, e.g., $E[T_{\alpha}]...
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Rescaling a Chebyshev polynomial

In this lecture: https://ee227c.github.io/notes/ee227c-lecture6.pdf the author states that he wants to create a polynomial $P_{k}(a)$ in terms of Chebyshev polynomials of the first kind with the ...
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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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6th Degree Polynomial and Chebyshev minmax

I have to find the 6th degree polynomial for the function $f(x)=xe^x$. After which the use of the Chebyshev min-max approach I have to use the list grade polynomial approach with respect to the fault ...
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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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Completeness relation for Chebyshev polynomials

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 ...
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What's the difference between $chebfun/@chebcolloc/$ and $chebfun/@valsDiscretization/$? [closed]

What's the difference between chebfun/@chebcolloc/ and chebfun/@valsDiscretization/? Are they using the same collocation method? ...
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Maths on the Paris Metro

My brother just saw this on the Paris Metro and asked what it is; unfortunately, it has defeated me. $$ {T_N} ^ {(2)} - {T_{N-1}} ^ {(1)} = {(LRC)}_{N - \frac{1}{2}}, $$ Does anyone know what it ...
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Integral with Chebyshev polynomials $ \int_{-1}^1 x^i T_n(x) dx $.

Is there any closed formula for such integral $$ \int_{-1}^1 x^i T_n(x) dx? $$ here $i$ is an integer numbers. For $n=2$ I have found that $$ \int_{-1}^1 x^i T_2(x) dx=\begin{cases}0,i=2k+1 \\ {\...
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some problem about chebyshev series

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 : $...
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Minimize integral of polynomial interpolation error

Given $\mathrm{f}\in C\left[-1,1\right]$ solve: $$ \min\int_{-1}^{1}\sqrt{\, 1 - x^{2}\,}\,\,\left\vert\,\mathrm{f}\left(x\right)-\mathrm{p}_n\left(x\right)\,\right\vert^{\,2}\,\mathrm{d}x $$ where $\...
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some problem for Chebyshev polynomials

I have a problem for Chebyshev polynomials. could you check whether my solution has fault or not show by induction that $T_n(x)= \frac{(x+\sqrt {x^2-1})^n+(x-\sqrt{x^2-1})^n}{2}$ pf: when $n=0$, ...
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Chebyshev polynomial

So I have a Chebyshev polynomial where I am trying to prove that $T_n(t)=\cos{(n \space \arccos{t})}, \space n=0,1,2...$ to form a system of orthogonal polynomials under the weighted inner product ...
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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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Chebyshev expansion of $\log(1 + x)$

I was reading a Wikipedia article on Chebyshev polynomials and got stuck in around the end of the article where the author takes advantage of orthogonality to compute the coefficients of the Chebyshev ...
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Does $\sum\limits_{i=1}^{n} f(\lambda_i) = \text{tr}(f(A))$ hold for any function $f$?

We know that given a $n$-by-$n$ matrix $A$, and its eigen values $\{\lambda_i\}_1^n$ its trace $\text{tr}(A)$. then the following holds: $$ \sum\limits_{i=1}^{n} \lambda_i = \text{tr}(A) $$ then ...