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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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Chebyshev polynomial property

I want to prove inequality (5.13) but I have a problem with (5.16). I have: $$ \sin(n\theta) = \sin\theta \cos(n-1)\theta + \sin(n-1)\theta \cos\theta = $$ $$ = \sin\theta \cos(n-1)\theta + \cos\...
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Chebyshev series on the complex plane

Denote $$T_n(x) := \cos(n \arccos(x)),\,\, n\in \mathbb{N}$$ the Chebyshev polynomials. Let $f$ be a continuous function on $[-1, 1].$ It is well known that $f$ can be written in its Chebyshev series ...
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googd reference for studying second kind Chebyshev wavelets

Can anyone recommend a good reference for second kind Chebyshev wavelets? I want to know how can generate them by a mother wavelet and that how they can form a (orthonormal) basis for the space $...
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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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Step to prove that $\cos (n \arccos (x))$ is a polynomial of $n$-th degree

I am confronted to the same problem stated in that question, namely to prove that cos(𝑛arccos(𝑥)) is a polynomial of 𝑛-th degree. However to begin with I don't understand how $$ \cos[n \...
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Integration of a Chebyshev series multiplied by an exponential function

I would like to evaluate the following integral: $$I(x_0)=\int_{-1}^1 \left(\sum_{k=0}^n a_k \, T_k(x)\right) \, \mathrm{e}^{b(x-x_0)}\,\mathrm{d}x$$ with $a_k$ some constant coefficients comming ...
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General Chebyshev approximation

I am having trouble understanding it, first of all, what is x? Are x's coefficients of this polynomial we are looking for? This would mean that the polynomial is of degree $n-1$ because it has n ...
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Maximum interpolation error in lagrange interpolation.

I have the following question: And the following Lagrange interpolation error bound: The way I have started to solve the problem is as follow. For me as a worst case is when all infinitely close to ...
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Chebyshev Polynomials of the Second Kind from Orthogonality

I am tasked with finding the degree 5 Chebyshev-II polynomial, using the fact that it's orthogonal to those preceding it w.r.t the Chebyshev-II inner product. I am told to use the normalisation that ...
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Map Chebyshev nodes on an arbitrary shape?

Is it possible to map Chebyshev nodes on an arbitrary shape (e.g. in 2D: on a triangle, in 3D: on a cone or pyramid)? The Chebyshev nodes will be used as interpolation points? Thanks.
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What are $\cos(\omega_k), -\sin(\omega_k)$ in Chebyshev filter design in matrix form?

What are $\cos(\omega_k), -\sin(\omega_k)$ in Chebyshev filter design in matrix form? The Chebyshev filter design problem "via SOCP" (https://en.wikipedia.org/wiki/Second-order_cone_programming) is ...
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Show that $\inf \{ \| f-P \|_{\infty}\mid P \in P_n \} \geq \delta_n$ for any decreasing sequence $\delta_n \to 0$

I'm trying to show that given any decreasing sequence $\delta_n \to 0$, we can find a continuous function $f: [-1,1] \to \mathbb{R}$ such that $$\inf\{\|f-P \|_{\infty}\mid P \text{ a polynomial of ...
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Vieta's Formula for Chebyshev basis

Let $p(x)=x^d+\sum_{i=0}^{d-1} a_ix^i$. Then Vieta's formula tells us that the $a_i$ can be expressed as signed elementary symmetric polynomials of the roots $\{\alpha_1,\ldots,\alpha_d\}$ of $p(x)$: $...
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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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approximating a univariate function $y=f(x)$ by roots of a bivariate polynomial

What is known about approximating a univariate monotone function $y=f(x)$ defined on $[0,1]$ or any finite domain by roots of a bivariate polynomial? For example a second order bivariate polynomial $...
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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 ...
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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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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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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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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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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 $[0,1]$....