Questions tagged [chebyshev-polynomials]
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.
267
questions
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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:
$$f(x)=f(0)+\frac{f^{(1)}(0)}{1!}(x-0)+\frac{f^{(2)}(0)}{2!}(x-0)^2+\frac{f^{(3)}(0)...
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votes
2answers
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Numerical evaluation of polynomials in Chebyshev basis
I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and ...
9
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1answer
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Polynomials with minimal variation and a fixed root---looking for a variant of Chebyshev polynomials (motivated by probability)
Recall that the Chebyshev polynomial $T_n(x)$ for a positive integer $n$ is, in a formal sense, the polynomial of degree $n$ that "varies the least" over an interval. Specifically, (a suitable scaling ...
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2answers
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How to best approximate higher-degree polynomial in space of lower-degree polynomials?
My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)?
Orginially, as the title of the post ...
8
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1answer
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How to get from Chebyshev to Ihara?
I have competing answers on my question about "Returning Paths on Cubic Graphs Without Backtracking". Assuming Chris is right the following should work. Up to one thing:
The number of returning paths ...
8
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Is there something like “associated” Chebyshev polynomials?
When I was experimenting with orthogonalization of polynomials
$$p_n(x)=\begin{cases}
1-x^n&\text{if }n\equiv0\; (\operatorname{mod}2),\\
x-x^n&\text{otherwise},
\end{cases}$$
i.e. simplest ...
7
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3answers
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Chebyshev's Theorem regarding real polynomials: Why do only the Chebyshev polynomials achieve equality in this inequality?
In the book Proofs from The Book by Aigner and Ziegler there is a proof of 'Chebyshev's Theorem' which states that if $p(x)$ is a real polynomial of degree n with leading coefficient $1$ then
$$ \...
7
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3answers
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Roots of the Chebyshev polynomials of the second kind.
It is known that the roots of the Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$ and they are simple (of multiplicity one). I have noticed that the roots ...
7
votes
1answer
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Prove that $\int_1^a \frac{T_n(x) T_n(x/a)}{\sqrt{a^2 - x^2} \sqrt{x^2 - 1^2}} \frac{a}{x} \mathrm{d}x = \frac{\pi}{2}$
In the paper, Representation of a Function by Its Line Integrals, with Some Radiological Applications, A. M. Cormack, Journal of Applied Physics 34, 2722 (1963), an integral identity is expressed ...
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Extending a Chebyshev-polynomial determinant identity
The following $n\times n$ determinant identity appears as eq. 19 on Mathworld's entry for the Chebyshev polynomials of the second kind:
$$U_n(x)=\det{A_n(x)}\equiv \begin{vmatrix}2 x& 1 & 0 ...
7
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1answer
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Combinatorial Interpretation of Graph Theoretical Relation Involving Chebyshev Polynomials
Given a graph $G$ and its adjacency matrix $A$. The $(i,j)$-th element of $A^r$ gives the number of ways to get from vertex $i$ to $j$ in $r$ steps (including backtracking).
Now, the number of ...
7
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1answer
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What is the connection between Taylor series and Chebyshev polynomials?
Can somebody help me find some historical references for the connection between Chebyshev polynomials and the Taylor series for sine and cosine functions? We know that Chebyshev polynomials are used ...
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1answer
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Definite integral involving Legendre polynomials with weight function $\sqrt{1-x^2}$
While investigating a problem in acoustic scattering in bounded domains, I encountered the following integral:
$$\int_{-1}^{1}\frac{\text{P}_n(x)\text{P}_m(x)}{\sqrt{1-x^2}}\mathrm{d}x$$
Where $\text{...
6
votes
1answer
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Show that $\inf \{ \| f-P \|_{\infty}\mid P \in P_n \} \geq \delta_n$ for any decreasing sequence $\delta_n \to 0$
I'm trying to show that given any decreasing sequence $\delta_n \to 0$, we can find a continuous function $f: [-1,1] \to \mathbb{R}$ such that $$\inf\{\|f-P \|_{\infty}\mid P \text{ a polynomial of ...
6
votes
1answer
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Polynomials with specified ranges in intervals
Say I have two finite intervals $[a,b],[c,d]\subsetneq\Bbb R$ where $a<b<c-1<c<d$ and $b-a=d-c=s<1$.
I want to find a polynomial $f \in \Bbb R[x]$ such that $$\forall x\in[a,b],\mbox{ }...
5
votes
3answers
452 views
Existence of polynomial such that $P_n(cos\theta)=cos(n\theta)$
Is there a way of proving existence of a polynomial $P_n(x)$ such that $\cos{(n\theta)}=P_n(\cos{\theta})$ without knowing the Chebyshev polynomials a priori?
5
votes
1answer
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Chebyshev expansion of $\log(1 + x)$
I was reading a Wikipedia article on Chebyshev polynomials and got stuck in around the end of the article where the author takes advantage of orthogonality to compute the coefficients of the Chebyshev ...
5
votes
2answers
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Trigonometric Identities for $\sin nx$ and $\cos nx $
These are generalizations of simple trigonometric identities for $\sin 2x$ and $\cos 2x$, but in general how can we prove them?
$$\sin nx =\sum_{k=1}^{\left\lceil\frac{n}{2}\right\rceil}(-1)^{k-1}\...
5
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0answers
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How do you define the sample points used for Chebyshev approximation/interpolation?
It appears there are somewhat conflicting definitions of the points used in Chebyshev interpolation. Wikipedia and Numerical Recipes define the $x_j^{(n)}$ sample points for $(n-1)^\text{th}$-order ...
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0answers
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What is this generalization of the Chebyshev polynomials?
For $\varepsilon>0$ consider the tridiagonal matrix
$$L_{\varepsilon}=\begin{bmatrix}
0 & 1 & \ & \ & \ & \ & \ & \ \\
1 & \varepsilon & 1 & \ & \ &...
4
votes
1answer
381 views
Chebyshev polynomials increase more quickly than any other polynomial outside $[-1,1]$
In Appendix C3 of Shewchuk's excellent notes on conjugate gradient, it is stated without proof that
Chebyshev polynomials... increase in magnitude more quickly outside the range $[-1,1]$ than any ...
4
votes
1answer
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Prove that $\prod\limits_{k=1}^{[n/2]} (3+2\cos\frac{2kπ}{n}) =F_n$ [closed]
$F_1=1; F_2=2; F_{n+1}=F_{n}+F_{n-1}$ (Fibonacci)
Prove that $\prod\limits_{k=1}^{[n/2]} \left(3+2\cos\dfrac{2kπ}{n}\right) =F_n$
please help me :(
4
votes
1answer
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How to Change the Interval of Interpolation from [-1,1] to [a,b] for Chebyshev Nodes
(According to this website:http://fac-staff.seattleu.edu/difranco/web/Math_371_W11/Files/Chebyshevnodes.pdf)
Between [-1,1], the Chebyshev Nodes are given as:
$x_k = \cos\Big((2k-1)\pi/2n)\Big), k=...
4
votes
1answer
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Is it possible to calculate the roots of the difference between successive terms of this polynomial series $\rm{P}_n (x)=x\rm{P}_{n-1}-r\rm{P}_{n-2}$
Consider the polynomial series defined by the following recursion formula:
$$
\begin{align}
&\mathrm{P}_0 = 1 \\
&\mathrm{P}_1 = x-r \\
&\mathrm{P}_n = x\mathrm{P}_{n-1} - r\mathrm{P}_{n-...
4
votes
1answer
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Estimating Spline curve by OLS. Is a good idea to fix the knots at Chebyshev sites?
I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or ...
4
votes
0answers
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Probabilties using Central Limit Theorem
Let n be an independent random variables and the number of orders in a 120 minute period.
Given that $\mu$ is 1.5 minutes and that $\sigma$ is 1 minute use the Central Limit Theorem to find the ...
4
votes
0answers
401 views
Complex Chebyshev Polynomials
Chebyshev Polynomials can be used to compute a very nearly minimax polynomial approximation of an analytic function on $[-1,1]$. Is there a complex analog that can compute a nearly minimax polynomial ...
3
votes
3answers
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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
$$T_n(x) = \frac{z_1^n + z_2^{-n} }{2}$$
...
3
votes
4answers
138 views
Extrema of Chebyshev polynomials (of the first kind)
I can hardly find a proof why the extrema of the Chebyshev polynomials are
$$
x_k=\cos(\frac{k}{n}\pi), k=1,...n
$$
and also why there are $n+1$ of them. The Chebyshev polynomials are here defined as ...
3
votes
2answers
436 views
show that nth Chebyshev polynomial is an nth order polynomial
Define the Chebyshev polynomial $T_n(x)=\cos(n\cos^{-1}(x)), n\geq 1, T_0=1)$.
Show that $T_n(x)$ is an nth order polynomial
This is my attempt, however I couldn't reduce it to a polynomial.
\...
3
votes
3answers
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Proving that $\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$
I recently saw on this site, the identity
$$\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$$
which I instantly wanted to prove.
I know that I can "reduce" the problem to the evaluation of $\...
3
votes
1answer
76 views
Palindromic combinations of Chebyshev Polynomials share common roots?
Suppose that the real polynomial below
$$p(x)=\sum_{k=0}^{2n}\alpha_{k}x^k$$
is a palindromic polynomial of even degree; that is, $p_{2n-k}=p_k$ for $0\leq k\leq 2n$ and $\alpha_0\neq 0$.
Is it ...
3
votes
1answer
395 views
On the extrema of Chebyshev polynomials of the second kind
I wish to prove that the magnitude of extreme values of $U_n(x)$, the Chebyshev polynomial of the second kind, is monotonically increasing on $[-1,1]$. By symmetry it suffices to prove it over $[0,1]$....
3
votes
1answer
59 views
Show that this integral is a multiple of a Chebyshev polynomial
Consider the function defined by the integral:
$$ f_n(x) = -\int_0^\pi \cos(n\theta)\log(|x-\cos\theta|)\,d\theta $$
I want to show that $f_n$ is a multiple of the nth Chebyshev polynomial on the ...
3
votes
4answers
159 views
What is the group-like structure on $x^2+y^2+z^2-2xyz=1$?
(Background: this is inspired by Chebyshev polynomials and expanding a function as a Chebyshev series.)
Solving for $ z $ gives
$$
z=xy \pm \sqrt{(1-x^2)(1-y^2)},
$$
where $-1\leq x,y \leq 1$. Now ...
3
votes
2answers
226 views
Limit superior of a sequence of oscillating functions related to Chebyshev polynomials
Let $n \in \mathbb N$ and consider the polynomial function $f_n \colon \mathbb R \to \mathbb R$ defined by
$$f_n(x) = \sum_{k=0}^n (-1)^k \binom {2n+1} {2k+1} (1 - x^2)^{n-k} x^{2k}$$
for any $x \in \...
3
votes
2answers
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Limit of Ratio of Chebyshev Polynomials
I have been trying to compute the limit
$$\lim_{n\to\infty}{{U_n(x)^2}\over{U_{n-1}(x)^2+U_n(x)^2}}$$
where $U_n(x)$ is the $n$-th Chebyshev polynomial of the second kind and $x\ge 1$.
Using software ...
3
votes
1answer
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What is a tensor-product Chebyshev grid?
What is the difference between "Chebyshev grid" and "tensor-product Chebyshev grid"?
Are they defined on a 2D vector?
3
votes
1answer
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The rate of convergence for polynomial interpolation of different functions
I have written some code in Matlab to polynomially interpolate functions with Chebyshev nodes using the Chebyshev base ( I calculate the coefficients with respect to the Chebyshev base and multiply ...
3
votes
1answer
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Chebyshev Polynomial Recurrence relation
the problem states: "Show that $\cos(n\theta)$ is a polynomial in $\cos(\theta).$" Now, using De Moivre's and Binomial theorems i get that
$$\cos(n\theta) = \sum_{k = 0, evens}^{n}\binom{n}{k}\cos^{n-...
3
votes
1answer
264 views
Question about chebyshev polynomial
Chebyshev polynomials are defined as such:
$$T_n(x)=\cos(n\arccos(x))$$
I'm asked to show that $\deg(T_j(x))=j$ and that $T_0,T_1,T_2,\ldots,T_n$ are an orthogonal basis of $\Bbb R_n[x]$.
I think I ...
3
votes
1answer
57 views
Recurrence relation for Chebyshev polynomials of the first kind
I tried to derive recurrence relation for Chebyshev polynomials from their generating function
$$\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_{n}(x)t^n.$$
I've differentiated both sides with respect to ...
3
votes
1answer
120 views
Jack d'Aurizio's exercise on Chebyshev polynomials [duplicate]
I am working through Jack D'Aurizio's “Superior Mathematics from an Elementary point of view”,
and I found (Lemma 61) the following lemma:
$\sum_{k=1}^{n-1}\frac{1}{\sin^2(\pi k/n)}=(n^2-1)/3$. He ...
3
votes
2answers
201 views
Uniqueness of minimal $\infty$-norm polynomial
From this proof it is clear to me that Chebyshev polynomial $\frac{1}{2^{n-1}} T_n(x)$ is minimum $\infty$-norm in $[-1,1]$ among the monic polynomials of degree $n$.
How to prove the uniqueness (if ...
3
votes
2answers
994 views
Chebyshev Interpolation and Expansion
I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials.
Pointwise Lagrange ...
3
votes
0answers
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$\cos\frac\pi{n}$ Analytic expression
I recently found out that $$\sin\frac\pi5=\frac12\sqrt{\frac{5-\sqrt5}2}$$
Which means that $$\cos\frac\pi5=\frac{1+\sqrt5}4$$
I also recently found that if $n\in\Bbb N$,
$$\sin nx=\sin x\,U_{n-1}(\...
3
votes
0answers
195 views
Why does this “incorrect” Chebyshev function approximation work better than the correct one?
I recently had the need to approximate this function
$$f\left(x\right)=\begin{cases}
\log\left(\frac{\pi+2\arcsin\left(x\right)}{\pi}\right), & x<0\\
-\log\left(2-\frac{\pi+2\arcsin\left(x\...
3
votes
0answers
150 views
From polynomials to Chebyshev polynomials
I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:
In order to obtain “strong” stability, we replace the condition ($2.7$) by
$$\left|\prod_{j=1}^N (1-\...
3
votes
0answers
89 views
Multivariate Appoximation
I have a mathematical model for a complex system which I would like to approximate it. My idea is to run this complex model once and produce some outputs, and then fit a polynomial for these outputs. ...
3
votes
0answers
151 views
Definite integral including the Chebyshev polynomial
I would like to know the proof of
$$ \int_a^b \frac{T_n(x/a)T_n(x/b)\, dx}{x(b^2-x^2)^{1/2}(x^2-a^2)^{1/2}}=\frac{\pi}{2 ab}, 0<a<b, n \in \Bbb N $$
where $T_n(x)$ is the Chebyshev polynomial of ...
