Questions tagged [chebyshev-polynomials]
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are two sequences of orthogonal polynomials which are related to de Moivre's formula. These polynomials are also known for their elegant Trigonometric properties, and can also be defined recursively. They are very helpful in Trigonometry, Complex Analysis, and other branches of Algebra.
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A book reference for the coefficients of the Chebyshev polynomial of the second kind
I am looking for a book reference for the following equation, presenting the coefficients of the Chebyshev polynomials of the second kind:
$$U_n(x)= \sum\limits_{j=0}^{n/2} (-1)^j \binom{n-j}{j}(2x)^{...
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Best uniform approximation of $x^{n+2}$ in $\mathbb{P_n}$
Let $n\geq 1$ be an integer and $f(x)=x^{n+2}$
for all $ x \in [−1, 1]$. Find the best uniform
approximation of $f$ in $\mathbb{P}_n$.
Attempt: Let's solve this first for $f(x)=x^{n+1}$ instead. ...
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Writing the $sin$ $cos$ power sum as a sum of multiple angles.
Writing the $sin$ $cos$ power sum as a sum of multiple angles.
Trying to answer the se question, which did not specify the multiple angle solutions, I started to look for a generalization and arrived ...
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Alternate sum of Chebyshev polynomials
The problem is
For all integer $n\ge1$,
\begin{align}\frac{(-1)^n}{2^{n-1}}\left(\frac12+\sum _{k=1}^n (-1)^{k} T_k(x)\right)&=\prod _{j=0}^{n-1} \left(x-\cos \left(\frac{\pi (2 j+1)}{2 n+1}\...
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A question about Chebyshev polynomials $T_n(x)$, $U_n(x)$, recurrence relations, and power of two $2^n$
I'm interested by the Chebyshev polynomials of the first kind $T_n(x)$ and of the second kind $U_n(x)$, especially $T_n(17)$ and $U_n(17)$.
The recurrence relation of $T_n(17)$ can be written as $a_{n}...
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Best uniform approximation of complex exponential function $e^z$ over unit disc in complex plane
It is known that the best uniform approximation for a real function defined in interval $[1,-1]$ is via the Chebyshev polynomials. ([see optimal polynomials])1. Such polynomials are also called min-...
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Compute the correction of a Chebyshev approximation using the Clenshaw summation formula
Assume you have a Chebyshev approximation of a function $f(x)$ evaluated using the Clenshaw summation method, up to polynomial order $N$:
$$
f(x) = \sum_{k=0}^{N-1} a_k T_k(x) = (a_0 - y_2)T_0(x) + ...
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Chebyshev polynomials orthogonal with respect to different weight function?
The following exercise appears in Ridgway Scott's Numerical Analysis:
Where $\omega_n(x)$ is the Chebyshev Polynomial of the first kind, that is $$\omega_{n+1}(x)=2^{-n}\cos((n+1)\cos^{-1}(x)$$
I ...
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dependency on length of interval in Chebyshev coefficients
Consider a function $g : [-1,1] \to \mathbb{R}$ and $c_n$ denotes the $nth$ Chebyshev coefficient of the function $g$. Moreover, $$c_n = \frac{2}{\pi}\int_{-1}^{1} T_n(x) g(x) \frac{1}{\sqrt{1-x^2}} ...
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Implementation reciprocal of a floating point number using Chebyshev approximation in CKKS
I am trying to obtain the reciprocal of a floating point value $x$ using the Chebyshev approximation, where $x$ is mostly in the order of $10^3$ to $10^5$. Subsequently, I am trying to implement that ...
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Closed form for infinite sum involving Chebyshev polynomials
There exists a generating function for the Chebyshev polynomials in the following form:
$$\sum\limits_{n=1}^{\infty}T_{n}(x) \frac{t^n}{n} = \ln\left( \frac{1}{\sqrt{ 1 - 2tx + t^2 }}\right)$$
...
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Roots and extrema of the polynomial $P_n(x)=\sum_{k=0}^n\binom{n+k}{2k}(-x)^k$.
Answering a recent question I came across the family of polynomials:
$$P_n(x)=\sum_{k=0}^n\binom{n+k}{2k}(-x)^k$$
with numerical evidence of the following interesting properties:
$P_n(2)=\begin{cases}...
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Bounds of polynomial approximation of a function of many variables using Jackson inequality
There is an approximation of a multivariate function by a Chebyshev polynomial of degree n. One needs to understand how the approximation error behaves depending on the degree of the polynomial or ...
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Largest root of a linear combination of Chebyshev polynomials
I wonder if we can say something about roots of a linear combination of Chebyshev polynomials of the first kind. I have an example in my hand: $$(m+1)T_n(x)+(m-3)T_{n-2}(x)=0$$ for some $m>0$. I ...
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Construct a manufactured solution of Poisson's equation with Chebyshev/Fourier expansions
I am solving a nonlinear Poisson's equation numerically using a mixed Chebyshev/Fourier spectral methods. Thus, assuming $x$ is periodic and $y$ is nonperiodic. I am trying to test my current ...
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Does Lagrange interpolation at Chebyshev points solve the Runge phenomenon?
I recently came across the concept of the Runge phenomenon while studying numerical methods for special functions in the book "Numerical Methods for Special Functions" by Amparo Gil, ...
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Formula for $T_k(x + 1)$ where $T_k$ is $k$-th Chebyshev polynomial of the first kind
Let $T_k(x)$ be the Chebyshev polynomial of the first kind of order $k$, i.e., the polynomial given by the recurrence relation
\begin{align}
T_0(x)=1, \quad T_1(x)=x, \quad T_k(x)=2xT_{k-1}(x)-T_{k-2}(...
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Density of Chebyshev nodes
While reading some notes, I came across the following statement:
``Chebyshev points have density $\mu(x) = \frac{N}{\pi\sqrt{1-x^2}}$".
I would like to understand where this formula comes from. ...
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Prove $\prod_{u=0}^{v-1}y\cos\frac{u\pi}v-x\sin\frac{u\pi}v=(-2)^{-v+1}\sum_{w=0}^{\lfloor\frac{v-1}2\rfloor}(-1)^w\binom{v}{2w+1}x^{v-2w-1}y^{2w+1}$ [closed]
Prove the following trigonometric equation
$$\prod_{u=0}^{v-1}y\cos{\frac{u\pi}{v}}-x\sin{\frac{u\pi}{v}}=(-2)^{-v+1}\sum_{w=0}^{\lfloor\frac{v-1}{2}\rfloor}(-1)^{w}\binom{v}{2w+1}x^{v-2w-1}y^{2w+1}$$
...
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Chebyshev approximation for bivariate function
I read the paper.
I am a litte bit confused regarding formulation of Chebyshev approximation for bivariate function(See photo). There is only one integral over variable x. Should it be in formula one ...
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Chebyshev differential equations
Consider the Chebyshev polynomial of the first kind
$$ (1-x^2)y'' - xy' + n^2y = 0 , n \in \mathbb{N}. $$
Use the substitution $ x=\cos\theta $ and show that the transformed ODE has solutions $y_1 = \...
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Convergence rate for Chebyshev polynomials to approximate $\text{erf}(x)$ on a subset of $\mathbb{R}$
Let $[-\alpha, \alpha] \subset \mathbb{R}$, and let
\begin{equation}
\text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}dt.
\end{equation}
given projections of $\text{erf}(x)$ onto the first $k$ ...
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Finding the Chebyshev polynomials $T_n$ by elementary means
Suppose that one person called the Student—virtually, an advanced schoolchild—obtained a tip that the Chebyshev polynomials of the first kind exist and unique for each $n$. By the Chebyshev polynomial ...
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What should the $\mathfrak{sl}(3)$ Chebyshev polynomials be?
Consider $\mathfrak{sl}_2$ and its fundamental weight $\lambda_1$. The character of the simple representation $L(n\lambda_1)$ with highest weight $n\lambda_1$ is given by a polynomial in $x=\mathrm{ch}...
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How is the Chebyshev polynomial approximation defined when using the spectral method to numerically solve PDEs?
I am studying spectral methods for numerical solutions for PDEs. Currently, I am on a chapter that explains how to use Chebyshev polynomials to solve non-periodic boundary value problems.
I understand ...
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How to derive the sum $ \sum_{k=1}^{n-1} \frac{1}{\cosh^2\left(\frac{\pi k}{n}\right)}$
\begin{align*}
\sum_{k=1}^{n-1} \frac{1}{\cosh^2\left(\frac{\pi k}{n}\right)}\end{align*}
I tried to solve with mathematica that shows
Does anyone know how to derive this and does it is possible for ...
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Coefficient of $x^n$ in Legendre series expansion
Suppose we are approximating a function $f$ with a Legendre series of order $N$, namely
$$
f(x) \approx \sum_{n=0}^N c_n P_n(x) \equiv f_N(x)
$$
where $P_n(x)$ is the $n^{th}$ Legendre polynomial and
$...
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Chebyhev polynomials and Primality Testing
It is a well known Theorem that an odd positive integer $n$ is prime if and only if
$T_{n}(x) \equiv x^n \pmod{n}$, where $T_{n}(x)$ is the $n^{th}$ Chebyshev polynomial of the first kind.
Do we ...
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Solving product of two cosine terms [closed]
I have an equation of $\cos Ax \cos Bx = c$ where $A$,$B$ and $c$ are known constants - how to solve for unknown $x$?
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Complex argument in Chebyshev polynomials of second kind?
I am looking at Chebyshev polynomials of second kind in order to characterize the spectra of $2$-Toeplitz perturbed matrices (I am not a mathematician myself, just a control theoretician). In all the ...
