All Questions
Tagged with chebyshev-polynomials real-analysis
35
questions
6
votes
2
answers
251
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 ...
- 543
0
votes
0
answers
158
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
$$
\sum a_nT_n(x) \quad \text{with} \quad a_n = \int_{-1}^{1}T_n(x)f(x)/\sqrt{1-x^...
- 499
2
votes
0
answers
139
views
Approximating Chebyshev Polynomials by Truncated Fourier Series
It is known (see e.g. https://www.math.ucdavis.edu/~bremer/classes/fall2018/MAT128a/lecture9.pdf) that any continuous function $f: [-1,1] \to \mathbb{C}$ admits a Chebyshev expansion
$$
f(x) = \sum_{n=...
- 191
3
votes
1
answer
157
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Is there any subsequence of the sequence $(\frac{\cos(\alpha - n \beta) - \lambda \cos(\alpha + n \beta)}{ \cos^n(\beta) })$ that converges?
Let's take $\alpha$ and $\beta$ two reals in $(0, \frac{\pi}{2})$.
Let's take $\lambda \in (0,1)$.
Let's define the sequence $(u_n)$ as follows:
\begin{eqnarray}
u_n & = & \frac{\cos(\alpha - ...
- 71
1
vote
1
answer
220
views
Relation between two extremal properties of Chebyshev polynomials
The Chebyshev polynomials $T_n$ have the following “minimal ∞-norm” property, also known as “Chebyshev's theorem”:
(A) Let $P$ be an $n$-the degree polynomial with leading coefficient $2^{n-1}$. Then
...
- 117k
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
0
answers
58
views
Fractional exponent elementary symmetric polynomials.
I am wondering if there is any literature on relations between fractional power symmetric polynomials. For a particular example, with the variables $\textbf{x} = (x_1,x_2,\dots x_n),$, can we ...
- 14.1k
-2
votes
2
answers
153
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Chebyshev polynomials semigroup property $T_n \circ T_m = T_{nm}$
Consider set of Chebyshev polynomials $T_n(x):\mathbb{R} \to \mathbb{R}$ given by formula
$$
T_n(\cos(x)) = \cos(nx)
$$
I am interested in elegant way to show that Chebyshev polynomials form a ...
- 927
0
votes
1
answer
82
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How we generalize the cartesian form of epicycloids?
I have the following parametric form of epicycloids:
$x(t)=\frac{a\cdot\cos t+\cos(a\cdot t)}{1+a}$
$y(t)=\frac{a\cdot\sin t+\sin(a\cdot t)}{1+a}$
where $a=2,3,4,\ldots$ is a variable that ...
user736865
2
votes
1
answer
1k
views
Prove the orthogonality relation of Chebyshev polynomials of the first kind
The Chebyshev polynomials of the first kind are obtained from the recurrence relation
$$\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x)~.\end{aligned}$$
Prove ...
- 7,880
0
votes
0
answers
64
views
Chebyshev expansion of $f(x)=\frac{1}{1+(x-s)^2}$
The Chebyshev polynomials of the first kind are obtained from the recurrence relation
$$\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x)~.\end{aligned}$$
I ...
- 7,880
0
votes
1
answer
43
views
Can we define the generating function for all $x$ and all $t$
The Chebyshev polynomials of second kind are defined for any $x \in \Bbb R$ (or
even $x \in \Bbb C$), e.g. via the recurrence relation
$$
U_0(x) = 1 \\
U_1(x) = 2x \\
U_{n+1}(x) = 2x U_n(x) - U_{...
- 3,854
1
vote
0
answers
125
views
Show that the lowest-norm monic polynomial is of the form $\frac{(b-a)^n}{2^n}\frac{1}{2^n}T_n\left(\frac{2}{b-a}x-\frac{b+a}{b-a}\right)$
Let $T_n\in P_n[-1,1]$ the n-th Chebyshev polynomial. Show that the lowest-norm monic polynomial in $P_n[a,b]$ is of the form $$\frac{(b-a)^n}{2^n}\frac{1}{2^n}T_n\left(\frac{2}{b-a}x-\frac{b+a}{b-a}\...
- 1,486
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
1
vote
0
answers
36
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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 ...
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