All Questions
Tagged with chebyshev-polynomials polynomials
64
questions
4
votes
2
answers
73
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Roots and extrema of the polynomial $P_n(x)=\sum_{k=0}^n\binom{n+k}{2k}(-x)^k$.
Answering a recent question I came across the family of polynomials:
$$P_n(x)=\sum_{k=0}^n\binom{n+k}{2k}(-x)^k$$
with numerical evidence of the following interesting properties:
$P_n(2)=\begin{cases}...
- 26.7k
0
votes
1
answer
68
views
Formula for $T_k(x + 1)$ where $T_k$ is $k$-th Chebyshev polynomial of the first kind
Let $T_k(x)$ be the Chebyshev polynomial of the first kind of order $k$, i.e., the polynomial given by the recurrence relation
\begin{align}
T_0(x)=1, \quad T_1(x)=x, \quad T_k(x)=2xT_{k-1}(x)-T_{k-2}(...
- 1,450
2
votes
3
answers
127
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Finding the Chebyshev polynomials $T_n$ by elementary means
Suppose that one person called the Student—virtually, an advanced schoolchild—obtained a tip that the Chebyshev polynomials of the first kind exist and unique for each $n$. By the Chebyshev polynomial ...
- 1,470
0
votes
0
answers
65
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How is the Chebyshev polynomial approximation defined when using the spectral method to numerically solve PDEs?
I am studying spectral methods for numerical solutions for PDEs. Currently, I am on a chapter that explains how to use Chebyshev polynomials to solve non-periodic boundary value problems.
I understand ...
15
votes
1
answer
452
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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 ...
- 2,279
3
votes
3
answers
101
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$T_n(x)^2 + T_n(y)^2-1$ is divisible by $x^2 + y^2-1$ where $T_n(x)$ is the Chebyshev polynomial, and $n$ odd
Found this interesting problem online:
Let $T_n(x)$ be the $n$-th Chebyshev polynomial. Show that for $n$ odd the polynomial
$$T_{n}(x)^2 + T_n(y)^2-1$$
is divisible by $x^2 + y^2-1$.
Notes:
It ...
- 55k
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
2
votes
0
answers
60
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Product Formula for Real Cyclotomic Polynomials
Let $n$ be a natural number, $\zeta_{n}$ be a primitive $n^{th}$ root of unity and $\Phi_{n}(x)$ be the $n^{th}$ cyclotomic polynomial. Let $\Psi_{n}(x)$ be the $n^{th}$ real cyclotomic polynomial (...
- 1,625
1
vote
1
answer
155
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Show that for Chebyshev polynomials satisfy $(1-x^2)T'_k + kxT_k -kT_{k-1} = 0$
$T_n(x)$ is the Chebyshev polynomials of the first kind. I was wondering how to show the following recurrence relationship:
$$ (1-x^2)T'_k + kxT_k -kT_{k-1} = 0$$
This recurrence form also shows up as ...
- 553
1
vote
0
answers
200
views
Chebyshev Polynomial of degree 3
This is the question:
(a) Given a function $f(x)=e^{-(a+1)x}$. Find the polynomial of degree $2$ , $p_{2}(x)$ such that (with $w(x)=1$):
$$||f-p_{2}||_{L^{2}(0,1)}=\min_{q \in P_{2}(0,1)}||f-q||_{L^{2}...
- 25
4
votes
5
answers
437
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Efficiently approximating the integral of $\operatorname{erf}(x y) \exp(-x^2)$
I need to efficiently approximate this integral which represents the Gaussian-weighted area of a right triangle whose three points are the origin plus two points that form a vertical "edge", ...
- 163
2
votes
1
answer
48
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How to show the approximation of $1-T_{1 / L}(1 / \delta)^{-2}$ when $L$ is large and $\delta$ is small?
I met this approximation problem in eq(2) of this paper, stating that the approximation is $\left(\frac{\log (2 / \delta)}{L}\right)^{2}$.
As stated in the question, $T_L(x)$ is the $L_{th}$ Chebyshev ...
- 67
4
votes
1
answer
71
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I am asking if this quantity has a name or it is just a real sequence.
Let us consider the Chebychev polynomial function $T_{n}(x)$ where $n$ is a fixed positive integer called the degree and $x$ is the real variable. Let us consider $x$ as a positive integer variable ...
- 3,854
3
votes
0
answers
257
views
Converting Chebyshev expansion into a regular polynomial
So I have a very very basic technique for converting shifted Chebyshev polynomials to regular polynomials but it seems to have a large issue with numerical stability and I don't completely understand ...
- 461
0
votes
1
answer
43
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Can we define the generating function for all $x$ and all $t$
The Chebyshev polynomials of second kind are defined for any $x \in \Bbb R$ (or
even $x \in \Bbb C$), e.g. via the recurrence relation
$$
U_0(x) = 1 \\
U_1(x) = 2x \\
U_{n+1}(x) = 2x U_n(x) - U_{...
- 3,854