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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 ...
Francesco Sollazzi's user avatar
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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. ...
miyagi_do's user avatar
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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 ...
Jakob's user avatar
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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}\...
hbghlyj's user avatar
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1 vote
1 answer
84 views

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}...
Aurel-BG's user avatar
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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-...
Manish Kumar's user avatar
2 votes
1 answer
50 views

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) + ...
LladOS's user avatar
  • 21
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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 ...
modz's user avatar
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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}} ...
Sam's user avatar
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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 ...
Sumana Bagchi's user avatar
3 votes
3 answers
299 views

Closed form for infinite sum involving Chebyshev polynomials

There exists a generating function for the Chebyshev polynomials in the following form: $$\sum\limits_{n=1}^{\infty}T_{n}(x) \frac{t^n}{n} = \ln\left( \frac{1}{\sqrt{ 1 - 2tx + t^2 }}\right)$$ ...
edrezen's user avatar
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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}...
user's user avatar
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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 ...
Masamune's user avatar
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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 ...
kswim's user avatar
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1 answer
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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 ...
Jamie 's user avatar
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135 views

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, ...
Swakshar Deb's user avatar
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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}(...
G. Gare's user avatar
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1 answer
79 views

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. ...
Okazaki's user avatar
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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 $$\prod_{u=0}^{v-1}y\cos{\frac{u\pi}{v}}-x\sin{\frac{u\pi}{v}}=(-2)^{-v+1}\sum_{w=0}^{\lfloor\frac{v-1}{2}\rfloor}(-1)^{w}\binom{v}{2w+1}x^{v-2w-1}y^{2w+1}$$ ...
Micheal Johnson's user avatar
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0 answers
45 views

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 ...
Masamune's user avatar
0 votes
0 answers
271 views

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 = \...
Jamied03's user avatar
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1 answer
89 views

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 $k$ ...
Cuhrazatee's user avatar
2 votes
3 answers
118 views

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 ...
Incnis Mrsi's user avatar
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5 votes
1 answer
143 views

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}...
Alvaro Martinez's user avatar
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0 answers
63 views

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 ...
FriendlyNeighborhoodEngineer's user avatar
1 vote
1 answer
164 views

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 ...
Martin.s's user avatar
0 votes
0 answers
73 views

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 $...
knuth's user avatar
  • 31
0 votes
1 answer
82 views

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 ...
Matrend's user avatar
1 vote
1 answer
52 views

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 $x$?
SathukaBootham's user avatar
0 votes
1 answer
330 views

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 ...
lyapunov00's user avatar
15 votes
1 answer
436 views

Show that $\max _{x \in[-1,1]}|P(x)| \geq \frac{1}{2^{n-1}}$ without Chebyshev polynomials

For every monic polynomial $P$ of degree $n$ (with leading coefficient 1), it is well-known that $$\max _{x \in[-1,1]}|P(x)| \geq \frac{1}{2^{n-1}}.$$ A standard proof uses Chebyshev polynomials. Is ...
Nathan Portland's user avatar
0 votes
1 answer
42 views

Show that for any positive integers $i$ and $j$ with $ i > j$, we have $T_i (x)T_j(x)= \frac{1}{2}[T_{i+j}(x)T_{i-j}(x)]$

Guys can you explain this demo to me step by step, I don't understand it at all. Show that for any positive integers $i$ and $j$ with $ i > j$, we have $$T_i (x)T_j(x)= \frac{1}{2}[T_{i+j}(x)T_{i-j}...
Del valle's user avatar
0 votes
0 answers
53 views

How to solve equation involving Chebyshev polynomial ratio?: $\frac{\cos\big(\frac{\cos^{-1}(ya)}{2N}\big)}{\cos\big(\frac{\cos^{-1}(y)}{2N}\big)}=x$

I have the following equation: involving the ratio of two Chebyshev polynomials: $$\frac{\cos\left(\frac{\cos^{-1}(ya)}{2N}\right)}{\cos\left(\frac{\cos^{-1}(y)}{2N}\right)}=x$$ (for some reason, I ...
RajaKrishnappa's user avatar
3 votes
2 answers
163 views

An integral identity: $\int_{0}^{\large\frac{π}{2}}\dfrac{\cos^{a}x\sin(a+1)x}{\sin x}\ \mathrm{d}x$

I recently came across a parametric definite integral about Chebyshev polynomials: $$ f(a)= \begin{aligned} \begin{gather*} \int_{0}^{\frac{\pi}{2}}\dfrac{\cos^{a}x\sin(a+1)x}{\sin x}\ \mathrm{d}x \...
Dylan Lee's user avatar
  • 151
4 votes
0 answers
171 views

Coefficients of Chebyshev polynomials

Not long ago, I derived the formula for Chebyshev polynomials $$T_{n}\left( x\right)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}{n \choose 2k}x^{n-2k}\left( x^2-1\right)^{k}$$ How to extract the ...
Fty56's user avatar
  • 149
0 votes
0 answers
48 views

Bounding the residual of Chebyshev approximations of the Heaviside function

I'm looking for a resource to find bounds for Chebyshev polynomial approximations of a given degree for the Heaviside function $$ H(x) = \begin{cases} 1 & x>0,\\ 0 & \text{else.} \end{cases}...
Cuhrazatee's user avatar
0 votes
0 answers
167 views

Inverse function of $U_{k-1}(\cos(\frac{\pi}{x}))$?

I'm trying to find the inverse function of $$U_{k-1}(\cos(\frac{\pi}{x}))=\sum_{n=0}^{\left\lfloor\frac{k-1}2\right\rfloor}\frac{(-1)^n \Gamma(k-n)}{n!\Gamma(k-2n)} \left(2\cos\left(\frac\pi x\right)\...
HarryXiro's user avatar
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1 vote
0 answers
41 views

Searching for weighted-$L^1$ summable orthonormal basis of $L^2(0,\infty)$

so I was working on something and bumped into the following question: Given some $a>0$, does there exist a complete orthonormal system $ (f_n)_{n \in \mathbb{N}} $ of $L^2(0,\infty)$ such that $\...
Dasi's user avatar
  • 256
0 votes
1 answer
54 views

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 $w$, ...
CACM6's user avatar
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2 votes
0 answers
49 views

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. ...
Teufel's user avatar
  • 91
3 votes
3 answers
100 views

$T_n(x)^2 + T_n(y)^2-1$ 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 ...
orangeskid's user avatar
  • 54.3k
1 vote
1 answer
172 views

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 ...
GSofer's user avatar
  • 4,323
-2 votes
1 answer
74 views

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 ...
Markova's user avatar
  • 23
1 vote
1 answer
110 views

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 ...
Brittany's user avatar
1 vote
0 answers
70 views

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 ...
Octavius's user avatar
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6 votes
0 answers
322 views

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 ...
Cuhrazatee's user avatar
6 votes
2 answers
245 views

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 ...
Mathews Boban's user avatar
3 votes
2 answers
145 views

Chebyshev polynomials to hypergeometric function?

I am trying to derive this hypergeometric version of the Chebyshev polynomials https://en.wikipedia.org/wiki/Chebyshev_polynomials#Explicit_expressions $$U_n(x) = \frac{\left (x+\sqrt{x^2-1} \right )^...
CarP24's user avatar
  • 310
1 vote
0 answers
18 views

Can the Chebyshev polynomials evaluated at a zero for index m, form a sequence of fixed length (in the index) with even parity?

For $n \in \mathbb{Z},n\geq 3$ find $x\in \mathbb{R}$ and index $m$ such that $T_m(x)=0$, $T_{m+j}(x)=T_{m-j}(x), \ j=1,\ldots, \lfloor \frac{n}{2} \rfloor -1$, $T_{m+j}(x) = T_{m+n+1-j}(x), \ j=1,\...
Brad Willms's user avatar
0 votes
0 answers
150 views

Question on convergence of Chebyshev series

I have written a script that plots some function and its truncated Chebyshev expansion in $[-1,1]$, which is given by $$ \sum a_nT_n(x) \quad \text{with} \quad a_n = \int_{-1}^{1}T_n(x)f(x)/\sqrt{1-x^...
user210089's user avatar

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