All Questions
Tagged with chebyshev-polynomials analysis
12
questions
3
votes
1
answer
54
views
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. ...
- 1,777
1
vote
1
answer
190
views
Prove the $k$-th power of the logistic map with parameter $\mu = 4$ has $2^k$ fixed points
I'm trying to solve an exercise in which I need to prove that the logistic map with parameter $\mu = 4$, $F_4:[0,1]\to[0,1]$, $F_4(x) = 4x(1-x)$, satisfies that for every positive integer $k$, $F_4^k$ ...
- 195
1
vote
0
answers
78
views
The envelope for the extremals of $\cos((n+1) \arccos x)-\cos((n-1) \arccos x)$ forms an ellipse.
The Chebyshev polynomial of the first kind is defined on $[-1, 1]$ by
$$T_n(x) = \cos(n \arccos x).$$
Prove that the envelope for the extremals of $T_{n+1}(x)-T_{n-1}(x)$ forms an ellipse.
The ...
- 919
2
votes
1
answer
57
views
Finding zeroeth coefficient of a Chebyshev polynomial expansion
Let $v_\theta = (\cos\theta,\sin\theta)$ be a unit vector in the plane. I have a kernel $p(\theta,\theta') = p(v_\theta\cdot v_{\theta'})$ that satisfies
$$\int_0^{2\pi} p(v_\theta\cdot v_{\theta'})\,...
- 1,088
2
votes
0
answers
77
views
One of Chebyshev's inequalities
How can I prove that this polynomial has at least n+1 zeroes? I have no idea.
- 492
1
vote
1
answer
178
views
Chebyshev polynomial property
I want to prove inequality (5.13) but I have a problem with (5.16). I have: $$ \sin(n\theta) = \sin\theta \cos(n-1)\theta + \sin(n-1)\theta \cos\theta = $$ $$ = \sin\theta \cos(n-1)\theta + \cos\...
- 27
2
votes
0
answers
319
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 ...
- 21
1
vote
1
answer
33
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\{||...
- 919
2
votes
0
answers
537
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 ...
- 3,198
7
votes
3
answers
3k
views
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 ...
- 1,849
2
votes
0
answers
125
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 ...
- 299
1
vote
1
answer
106
views
Coefficients of Chebychev Polynomials
Is there a known formula for the coefficient of x^k in the nth chebychev polynomial of the first kind?
- 11