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.
109
questions with no upvoted or accepted answers
8
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124 views
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 ...
6
votes
1answer
127 views
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 $$\forall x\in[a,b],\mbox{ }...
5
votes
0answers
223 views
How do you define the sample points used for Chebyshev approximation/interpolation?
It appears there are somewhat conflicting definitions of the points used in Chebyshev interpolation. Wikipedia and Numerical Recipes define the $x_j^{(n)}$ sample points for $(n-1)^\text{th}$-order ...
5
votes
0answers
75 views
What is this generalization of the Chebyshev polynomials?
For $\varepsilon>0$ consider the tridiagonal matrix
$$L_{\varepsilon}=\begin{bmatrix}
0 & 1 & \ & \ & \ & \ & \ & \ \\
1 & \varepsilon & 1 & \ & \ &...
4
votes
1answer
100 views
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
72 views
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
400 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
0answers
73 views
$\cos\frac\pi{n}$ 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 $$
where $T_n(x)$ is the Chebyshev polynomial of ...
3
votes
1answer
223 views
Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?
I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
2
votes
0answers
56 views
Eigenvalues of Jacobian of Mandelbulb “triplex” power formula
I'm trying to find a lower bound for the distance estimate of the Mandelbulb fractal, or at least justify why using the scalar-derivative for distance estimation is so effective. The Mandelbulb ...
2
votes
0answers
64 views
One of Chebyshev's inequalities
How can I prove that this polynomial has at least n+1 zeroes? I have no idea.
2
votes
1answer
174 views
Change of Interval for Chebyshev–Gauss quadrature
I am curretly working to numerically evaluate an integral of the form:
$$\int_{-1}^{1} f(x) \sqrt{1-x^2} dx$$
For this issue Gauss-Chebysehv integration of second kind seems ideal as it uses the ...
2
votes
1answer
53 views
Chebyshev Inequality - How is the following inferred ??
In chapter 3, Norm and Distance of Introduction to Applied Linear Algebra by Boyd, an example explaining the Chebyshev inequality for standard deviation is given as:
Consider a time series of return ...
2
votes
0answers
51 views
Vieta's Formula for Chebyshev basis
Let $p(x)=x^d+\sum_{i=0}^{d-1} a_ix^i$. Then Vieta's formula tells us that the $a_i$ can be expressed as signed elementary symmetric polynomials of the roots $\{\alpha_1,\ldots,\alpha_d\}$ of $p(x)$: $...
2
votes
0answers
179 views
Chebyshev coefficients- interpolation on [a,b]
My problem is to solve a second order differential equation given two (Dirichlet) boundary conditions.
$\frac{d^2y}{dx^2} = M/EI$
Both M and I are functions of x. Owing to complexity of the ...
2
votes
0answers
118 views
Minimal error chebyshev interpolation
Let's say the n-degree Chebyshev polynomials :
$$ T_{n} (x)=\cos(n\arccos(x))$$
Make a polynomial such that:
$$\mid y- P (x) \mid$$
be minimal, using the first three Chebyshev polynomials for the ...
2
votes
0answers
58 views
Is there a general form of Chebyshev expansion coefficients for Gaussian distribution
$\newcommand{\chebyshevt}{\text{chebyshevt}}$
$\newcommand{\Norm}{\text{Norm}}$
I tried to calculate the coefficient for distribution $Norm(x, \mu, \sigma)$ via
$$\int_{-1}^{1} \chebyshevt(x, t) \...
2
votes
0answers
25 views
Numerical analysis: Chebyshev coefficient representation error.
I am unsure if numerical analysis questions are suitable for this forum, but I have nowhere else to ask, so if this question is inappropriate, tell me and I will delete it.
If $x_k$ are the Chebyshev ...
2
votes
0answers
172 views
Pythagorean-like equation for generalized hyperbolic function
Trig functions satisfy $\cos^2t+\sin^2t=1$, which is an expression of the Pythagorean theorem. Hyperbolic trig functions satisfy $\cosh^2t-\sinh^2t=1$ which may perhaps be viewed as a generalization. ...
2
votes
0answers
81 views
Demystifying math: how could someone come up with Chebychev polynomials?
I hope this question is allowed, I am interested how you think someone could come up with the Chebychev polynomials, where I refer to them in the sense that someone would be interested in the ...
2
votes
0answers
388 views
Generating Chebyshev polynomials by Gram-Schmidt
Given the definition of Chebyshev polynomials in this form:
$$T_n(x) = \cos(n\cos^{-1}x), n\ge 1, T_0=1$$
I want to show that using Gram-Schmidt procedure with set $\{1, x, x^2, \dots\}$ and weight ...
2
votes
0answers
615 views
How would you compute the eigenvalues of a finite difference operator?
Having read: What are eigenvalues of higher order finite differences matrices?
I am still unclear how you would do this for an arbitrary matrix:
$$A=\begin{pmatrix}
a&b&e&0&0&0&...
2
votes
0answers
144 views
Find the cubic near-minimax or Chebyshev approximation for $f(x)=\sin(x)$
Find the cubic near-minimax or Chebyshev approximation for $f(x) = \sin(x)$ on the interval $[0,\frac{\pi}{2}]$.
Attempt: The first four Chebyshev polynomials are
\begin{align} T_0(x)&=1,\\ ...
2
votes
0answers
48 views
Solution to an $n^{th}$ order polynomial equation as a series in $n$
If I have an $n^{th}$ order polynomial set equal to zero, is there some way I can invert it to get a series expansion in terms of $n$?
In particular I'm interested in solving the equation
$$ U_n(x) -...
2
votes
0answers
67 views
Constraints on a Chebyshev series representation of a CDF
My question is about deriving constraints for coefficients of a
Chebyshev series which represents a CDF.
Let $F(x)$ be the cumulative distribution function for $x\in [-1,1]$. Accordingly we know ...
2
votes
0answers
273 views
Multivariate Clenshaw Chebyshev Algorithm (downward recursion)
I have recently written a code where I use Clenshaw's summation formula with Chebyshev polynomials
$S(x)=\sum_{k=0}^nc_kT_k(x)=b_0+xb_1$
$T_{k+1}(x)=2xT_k(x)-T_{k-1}(x)$
$T_0(x)=1~~~ T_1(x)=x$
$b_{...
2
votes
0answers
452 views
Inner Product of Chebyshev polynomials of the second kind with $x$ as weighting
I have tried to solve the integral
$${\int_0^1 U_n (x) U_m (x)x dx },$$
where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind.
Solving the integration and checking the result, I ...
2
votes
0answers
157 views
Size of coefficients of polynomials that satisfy a Chebyshev-like extremal property
The famous Chebyshev polynomials satisfy many extremal properties. One of these is that they attain the largest possible derivative over the interval [-1,1] among polynomials whose absolute value over ...
2
votes
0answers
249 views
How to find coefficients in a multivariate Chebyshev polynomial approximation
How do perform a multivariate Chebyshev approximation?
Let \begin{align}
\vec{x} & = x_{0}, x_{1}, ... , x_{n},\\
\vec{a} & = a_{0}, a_{1}, ... , a_{n},\\
\vec{b} & = b_{0}, b_{1}, ... , ...
2
votes
0answers
424 views
Sampling a Chebyshev polynomial with the discrete cosine transform
I have a Chebyshev polynomial $f$ of degree $n$
in point-value form
\begin{align}
f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\
x_i &= \cos\left( \frac{i \pi}{n} \right), ...
2
votes
0answers
256 views
Chebyshev polynomials approximation - Is there a way to generalize this
In an exam I was given this question:
let $f(x)=x^3$. We want to find the best linear approximation (best in the sense that the maximal error is minimized) of $f$ in the interval $[-1,1]$ using ...
2
votes
0answers
107 views
how to prove this curious identity with the Chebyshev polinomials
we defined the Tm like this (where Tm are the Chebyshev polinomials)
Then I showed this:
And now I have no idea how to proove this:
I also have to make the remark that I also proved that the ...
2
votes
0answers
31 views
Give bounds for degree of “decreasing” polynomial
Let $p$ be a polynomial of minimal degree to which the following is true:
$p(0) > \sum\limits_{i=1}^n \lvert p(i) \rvert + \sum\limits_{i=1}^m \lvert p(-i) \rvert$
Give upper and lower bounds for $...
1
vote
0answers
51 views
Show that the lowest-norm monic polynomial is of the form $\frac{(b-a)^n}{2^n}\frac{1}{2^n}T_n\left(\frac{2}{b-a}x-\frac{b+a}{b-a}\right)$
Let $T_n\in P_n[-1,1]$ the n-th Chebyshev polynomial. Show that the lowest-norm monic polynomial in $P_n[a,b]$ is of the form $$\frac{(b-a)^n}{2^n}\frac{1}{2^n}T_n\left(\frac{2}{b-a}x-\frac{b+a}{b-a}\...
1
vote
0answers
32 views
Chebyshev interpolating polynomial theorem
Could anyone explain this Chebyshev interpolating polynomial theorem 3.5 ?
1
vote
1answer
46 views
How would one go about computing a Chebyshev-type quadrature problem with general integration limits of [a,b] instead of [-1,1]?
More specifically, a problem of the form
$\int_a^b\frac{f(x)}{\sqrt{1-x^2}}dx = \sum_{i=1}^{N}w_if(x_i)$,
where $a,b \in [-1,1]$, $w_i$ are the weights, and $x_i$ are the abscissa. A quick search ...
1
vote
0answers
24 views
Chebyshev polynom problem.
Consider $P_n = \{Q_n(x) : \deg Q_n = n; \|{Q_n}\| = \max_{[a,b]}|Q_n(x)| = M >0\}$.
Now consider $\bar{T}(x) = M T_n(\frac{2x - (b+a)}{b-a}) - $ Chebyshev polynomial (normed on space $P_n)$.
We ...
1
vote
0answers
33 views
Chebyshev Approximation : concentrating nodes on specific intervals
I have a smooth function :
$$\frac{1}{2\pi} \sin(2\pi x )~,$$
that I would like to approximate on an interval $[-a, b]$. However I already know that all the inputs are guarantied to be close to an ...
1
vote
0answers
72 views
Chebyshev series on the complex plane
Denote $$T_n(x) := \cos(n \arccos(x)),\,\, n\in \mathbb{N}$$
the Chebyshev polynomials. Let $f$ be a continuous function on $[-1, 1].$ It is well known that $f$ can be written in its Chebyshev series ...
1
vote
0answers
19 views
googd reference for studying second kind Chebyshev wavelets
Can anyone recommend a good reference for second kind Chebyshev wavelets?
I want to know how can generate them by a mother wavelet and that how they can form a (orthonormal) basis for the space $...
1
vote
0answers
165 views
Chebyshev Polynomials of the Second Kind from Orthogonality
I am tasked with finding the degree 5 Chebyshev-II polynomial, using the fact that it's orthogonal to those preceding it w.r.t the Chebyshev-II inner product. I am told to use the normalisation that ...
1
vote
0answers
67 views
Map Chebyshev nodes on an arbitrary shape?
Is it possible to map Chebyshev nodes on an arbitrary shape (e.g. in 2D: on a triangle, in 3D: on a cone or pyramid)? The Chebyshev nodes will be used as interpolation points?
Thanks.
1
vote
0answers
41 views
Is the following matrix defined by the roots of Chebyshev polynomial invertible?
Let $x_0, \dots , x_n$ the roots of the Chebyshev polynomial, $T_{n+1}(x)$.
We define:
$\begin{pmatrix}
\frac{1}{\sqrt2}T_0(x_0) & \cdots & \frac{1}{\sqrt2}T_0(x_0) \\
T_1(x_0) & \...
1
vote
0answers
120 views
Use 1-degree Chebyshev polynom to approximate $\cos(x)$ and calculate the error
The task is to give for $\cos(x)$ the nodes of the interpolation polynom of degree 1 that approximates the function on $[-\pi,\pi]$ the best as well as the related error.
I want to solve this task ...
1
vote
0answers
216 views
Completeness relation for Chebyshev polynomials
The Chebyshev polynomials of the first kind $T_n(x)$ are known to form a complete orthogonal basis for functions on $[-1,1]$. I was looking for a proof of the completeness part, without any luck, when ...
1
vote
1answer
31 views
some problem about chebyshev series
Suppose that $f \in C[-1,1]$ has a chebyshev series $\sum_{n=1}^{\infty}a_nT_n$
(b) show that $E_n(T_{n+1})=1$
(c) show that $|E_n(f)-|a_{n+1}|| \le \sum_{k=n+2}^{\infty}|a_k|$
cf : $E_n(f)= \inf\{||...
