# 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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### Multiple angle formulas for tangent

There are direct multiple angle formulas for sine and cosine, in terms of Chebyshev polynomials. For example, the cosine of $n\theta$ is $$\cos{n\theta}=T_n(\cos\theta)$$ Where $T_n$ is the n-th ...
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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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### 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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### 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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### why no initial conditions are required in the differential equation/eigenfunctions problem of orthogonal polynomials?

in section 4.2 of the book "Special functions, a graduate text" says the following: We return to the three cases corresponding to the classical polynomials, with interval $I$, weight $w$, ...
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### Determine the minimal value of $\frac{3a}{b+c}+\frac{4b}{c+a}+\frac{5c}{a+b}$ [duplicate]

Let $a$, $b$, $c$ be positive real numbers. Determine the minimal value of $$\frac{3a}{b+c}+\frac{4b}{c+a}+\frac{5c}{a+b}.$$ I have solved this problem using Chebyshev's and Nesbitt's inequalities. ...
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### $T_n(x)^2 + T_n(y)^2-1$ is divisible by $x^2 + y^2-1$ where $T_n(x)$ is the Chebyshev polynomial, and $n$ odd

Found this interesting problem online: Let $T_n(x)$ be the $n$-th Chebyshev polynomial. Show that for $n$ odd the polynomial $$T_{n}(x)^2 + T_n(y)^2-1$$ is divisible by $x^2 + y^2-1$. Notes: It ...
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### Generalized Chebyshev Polynomials and trigonometric identities

The usual Chebyshev polynomials of the first kind are defined by the recursive relation: $$T_0(x)=1,\\ T_1(x)=x,\\ T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$$ One can use the same recursive relation to define ...
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### I am having trouble getting started with this problem. I am not sure how to approach it, if someone is willing to give me a hint?

Show that the Chebyshev-Lobatto points $(\eta_k)^n = \cos\left(\dfrac{kpi}{n}\right)$ are the zeros of the polynomial $$(1-x^2)\dfrac{\sin(n\theta)}{\sin\theta}$$ where $x = \cos\theta$. The ...
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### Approximating $e^{-2x}$ using a linear combination of Chebyshev polynomials

I've been asked to find the coefficients $c_0, c_1, c_2, c_3$ for $P_4(x) = c_0 \, T_0(x) + c_1 \, T_1(x) + c_2 \, T_2(x) +c_3 \, T_3(x)$ given $f(x) = e^{-2x}$, where $T_n(x)$ are Chebyshev ...
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### Differentiation matrix for Chebychev spectral-method

I am currently familiarising myself with spectral methods for solving differential equations. Now I am confused when it comes to calculating derivatives. Let $f(x)$ be a function and let $f_N(x)$ be ...
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### A stereographic projection for the Chebyshev polynomials

This question may be too vague for the MSE crowd, if so, please feel free to ask clarifying questions or just remove. The Chebyshev polynomials are a family of orthogonal polynomials typically defined ...
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### Polynomial that grows faster than any other polynomial outside $[−1,1]^n$

Consider this statement: "Chebyshev polynomials increase in magnitude more quickly outside the range $[−1,1]$ than any other polynomial that is restricted to have magnitude no greater than one ...
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