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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