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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.

The Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula.

There are two kinds of these polynomials. The first kind $T_n$ is defined by the recurrence $$\begin{align} T_0(x)&=1\\ T_1(x)&=x\\ T_{n+1}(x)&=2xT_n(x)-T_{n-1}(x) \end{align}$$ The second kind $U_n$ is defined by the same recurrence, but with $U_1(x)=2x$.

These polynomials also satisfy the trigonometric identities $$T_n(\cos\theta)=\cos(n\theta)\qquad U_n(\cos\theta)\sin\theta=\sin(n+1)\theta.$$

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Generalizing two infinite products for $\operatorname{sinc}(x)$ and their 'dual' infinite product

$\newcommand{\sinc}{\operatorname{sinc}}$ Throughout, let $m,k$ be positive integers, $x>0$ a real number, and denote $\sinc(z)=\sin(z)/z$ with $\sinc(0)=1$. A famous result of Euler gives $\sinc(x)$ as an infinite…
Integrand
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How to use Chebyshev Polynomials to approximate $\sin(x)$ and $\cos(x)$ within the interval $[−π,π]$?

I have approximated $\sin(x)$ and $\cos (x)$ using the Taylor Series (Maclaurin Series) with the following results: $$f(x)=f(0)+\frac{f^{(1)}(0)}{1!}(x-0)+\frac{f^{(2)}(0)}{2!}(x-0)^2+\frac{f^{(3)}(0)}{3!}(x-0)^3+\cdots$$ $$\begin{align}\implies…
David Lin
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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 there a proof which does not use Chebyshev…
Nathan Portland
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Numerical evaluation of polynomials in Chebyshev basis

I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and -1, where Chebyshev polynomials change rapidly,…
Omicron_Persei_11
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Chebyshev's Theorem regarding real polynomials: Why do only the Chebyshev polynomials achieve equality in this inequality?

In the book Proofs from The Book by Aigner and Ziegler there is a proof of 'Chebyshev's Theorem' which states that if $p(x)$ is a real polynomial of degree n with leading coefficient $1$ then $$ \max_{-1 \leq x \leq 1} |p(x)| \geq…
Brusko651
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How to best approximate higher-degree polynomial in space of lower-degree polynomials?

My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)? Orginially, as the title of the post suggests, I'm asking the general problem: given an…
user33869
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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 of) $T_n(x)$ is the monic polynomial $p$ of…
Noah Stephens-Davidowitz
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How to get from Chebyshev to Ihara?

I have competing answers on my question about "Returning Paths on Cubic Graphs Without Backtracking". Assuming Chris is right the following should work. Up to one thing: The number of returning paths on 3-regular graphs of length $r$ without…
draks ...
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Show that $\prod_{k=1}^{n}\sqrt{1+8\sin^2(k\pi/n)}=2^n-1$ for all positive integers $n$.

Let $P(n)=\prod\limits_{k=1}^{n}\sqrt{1+8\sin^2\left(\frac{k\pi}{n}\right)}$ Show that $P(n)=2^n-1$ for all positive integers $n$. Context You may have heard of the following remarkable fact about the unit circle: If $n$ equally spaced points are…
Dan
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Integrating Chebyshev polynomial of the first kind

I'm trying to evaluate the integral of the Chebyshev polynomials of the first kind on the interval $-1 \leq x \leq 1 $ . My idea is to use the closed form $$T_n(x) = \frac{z_1^n + z_2^{-n} }{2}$$ where $z_1 = (x + \sqrt{x^2 - 1})$ and $z_2 = (x -…
M3rlyn
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Is there something like "associated" Chebyshev polynomials?

When I was experimenting with orthogonalization of polynomials $$p_n(x)=\begin{cases} 1-x^n&\text{if }n\equiv0\; (\operatorname{mod}2),\\ x-x^n&\text{otherwise}, \end{cases}$$ i.e. simplest binomials vanishing at $x=-1$ and $x=1$, with respect to…
Ruslan
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What is the connection between Taylor series and Chebyshev polynomials?

Can somebody help me find some historical references for the connection between Chebyshev polynomials and the Taylor series for sine and cosine functions? We know that Chebyshev polynomials are used to represent multiple angle identities for sine…
Bob Tivnan
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Existence of polynomial such that $P_n(\cos\theta)=\cos(n\theta)$

Is there a way of proving existence of a polynomial $P_n(x)$ such that $\cos{(n\theta)}=P_n(\cos{\theta})$ without knowing the Chebyshev polynomials a priori?
Kal S.
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Combinatorial Interpretation of Graph Theoretical Relation Involving Chebyshev Polynomials

Given a graph $G$ and its adjacency matrix $A$. The $(i,j)$-th element of $A^r$ gives the number of ways to get from vertex $i$ to $j$ in $r$ steps (including backtracking). Now, the number of reduced paths on cubic graphs of length $n$ (without…
draks ...
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Monte-Carlo integration

Let a function $f$ to be $x\in \left[a,b\right],\:0\le f\left(x\right)\le c$. We want to calculate the approximation of the definite integral of the function in the range $[a,b]$, we can suppose that the exact integral is very difficult to calculate…
user869856
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