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 a formula to derive the index of a cyclic series element $s_k$ with $s_{i+1} = p(s_i) \mod N$ with $p$ a Chebyshev polynomial modulo $N$

Chebyshev_polynomials (of first order) $T_n(x)$ are some special commutative polynomials which are equal to $T_n(x) = \cosh(n \operatorname{arcosh}(x))$ for $x>=1$. e.g. $$T_6(x) = 32x^6 - 48x^4 + ...
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Given a sequence $n_{i+1} = f(n_i)$ with $f$ a polynomial of degree m. Any way to get the index $i$ of a sequence element $v$?

A sequence $$n_{i+1} = f_m(n_i)$$ with $f_m$ a polynomial of degree $m$. With this also $n_i$ is a polynomial: $$n_i=\sum_{j=0}^{m\cdot i} a_jx^j$$ If $n_1$ has polynomial degree $m$ then the ...
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Change of Interval for Chebyshev–Gauss quadrature

I am curretly working to numerically evaluate an integral of the form: $$\int_{-1}^{1} f(x) \sqrt{1-x^2} dx$$ For this issue Gauss-Chebysehv integration of second kind seems ideal as it uses the ...
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Chebyshev polynom problem.

Consider $P_n = \{Q_n(x) : \deg Q_n = n; \|{Q_n}\| = \max_{[a,b]}|Q_n(x)| = M >0\}$. Now consider $\bar{T}(x) = M T_n(\frac{2x - (b+a)}{b-a}) - $ Chebyshev polynomial (normed on space $P_n)$. We ...
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Transforming a Power series to a Chebyshev expansion

I was wondering whether there is a simple alogorithm of taking a power series, $\sum\limits_{n=0}^\infty a_n x^n$, and rewriting it in the form of Chebyshev expansion, i.e: $$ \sum\limits_{n=0}^\...
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Finding a sequence of polynomials with certain properties

I know that the Chebyshev polynomials , $\{ T_n \}_{n=1}^\infty$, satisfy the property that: $$ \int_{-\omega}^\omega T_n(x/\omega)\cdot \frac{k\sqrt{\omega^2-x^2} }{2\pi(k^2-x^2)}dx=\begin{cases} -\...
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Proving that $\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$

I recently saw on this site, the identity $$\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$$ which I instantly wanted to prove. I know that I can "reduce" the problem to the evaluation of $\...
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Gauss-Chebyshev Integration on arbitrary intervals

So, I would like to do a numerical integration of something of the form: $\int_{-1}^1 dx f(x) \sqrt{1-x^2}$ I tried this using Gauss-Legendre only to find out that the last second factor keeps the ...
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Chebyshev Approximation : concentrating nodes on specific intervals

I have a smooth function : $$\frac{1}{2\pi} \sin(2\pi x )~,$$ that I would like to approximate on an interval $[-a, b]$. However I already know that all the inputs are guarantied to be close to an ...
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Chebyshev Inequality - How is the following inferred ??

In chapter 3, Norm and Distance of Introduction to Applied Linear Algebra by Boyd, an example explaining the Chebyshev inequality for standard deviation is given as: Consider a time series of return ...
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Give the upper limit of the difference between truncated power series and economized one (economized using Chebyshev polynomials).

The question is as follow: (a) The first three Chebyshev polynomials are: $$T_0=1$$ $$T_1 = x$$ $$T_2 = 2x^2-1$$ $$T_3 = 4x^3-3x$$ $$T_4 = 8x^4-8x^2+1$$ i) Economize the truncated power series: $$p(...
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Uniqueness of minimal $\infty$-norm polynomial

From this proof it is clear to me that Chebyshev polynomial $\frac{1}{2^{n-1}} T_n(x)$ is minimum $\infty$-norm in $[-1,1]$ among the monic polynomials of degree $n$. How to prove the uniqueness (if ...
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Chebyshev's series for square-root of x over $x\geq 0$.

I am looking for the derivation of Chebyshev's series expansion for $\sqrt{x}$ over $x\in \mathbb{R}^{+}$. I am little confused on how to go from $[-1,1]$ range of Chebyshev's polynomial range to $\...
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Proving that the only integer solution of $2U_{2M}(\sqrt{q}/2)=U_{2(M+1)}(\sqrt{q}/2)$ for $M\in\mathbb{Z}_{+}$ is $q=3$

I'd like to prove that the only integer solution of $2U_{2M}(\sqrt{q}/2)=U_{2(M+1)}(\sqrt{q}/2)$ for $M\in\mathbb{Z}_+$ is $q=3$ where $U$ is the Chebyshev polynomial of the second kind and I've taken ...
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Higher-order Derivatives of Legendre Polynomials at endpoints

I am looking for simple formulas for higher-order derivatives of Legendre polynomials $P^{(k)}_{n}(\pm 1)$. For the Chebyshev polynomials, there is a simple formula $$ \left. \frac{d^{p}T_n}{dx^p}\...
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Is it possible to solve tridiagonal Toeplitz matrix whose center element is different, using Chebyshev polynomial of the second kind?

I have a tridiagonal Toeplitz matrix whose first diagonal below main diagonal, and the first diagonal above the main diagonal have elements equal to $-1$ and the main diagonal elements are same ...
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Polynomials with minimal variation and a fixed root---looking for a variant of Chebyshev polynomials (motivated by probability)

Recall that the Chebyshev polynomial $T_n(x)$ for a positive integer $n$ is, in a formal sense, the polynomial of degree $n$ that "varies the least" over an interval. Specifically, (a suitable scaling ...
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How to get the basis of $\cos(nx)$ in Chebyshev polynomials.

Given $\cos(nx) = f_n(\cos x)$ for any positive integer $n$ and polynomial $f_n(x) = 2^{n-1}x^n + a_1 x^{n-1} + ... + a_{n-1}x + a_n$. Show if $n$ is odd, $a_n = 0$ and if $n$ is even, $a_n=(-1)^{\...
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How to find norm $||U_n||$ of Chebyshev polynomials of the second kind?

how to find norm $||U_n||$ and the values $U_n(\pm)$ of the Chebyshev polynomials of the second kind?
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Showing a Chebyshev set

I want to show that $\{1,e^{ix},...,e^{(n-1)x} \}$ is a Chebyshev Set on $(0,2\pi]$. Now I know that $\{1,x,...,x^n \}$ is one and that $e^{ix}$ is injective on $(0,2\pi]$. But how do I show that if I ...
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Chebyshev Filter Low Pass Conceptual Question

If I have a 2D signal (say a 2D image) thats defined on $[-1,1]^2$. I sample the 2D signal on discrete Chebyshev Points (Chebyshev-Legendre Points), say there is 60 grid points per side. The ...
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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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66 views

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 ...