Questions tagged [chebyshev-polynomials]
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are two sequences of orthogonal polynomials which are related to de Moivre's formula. These polynomials are also known for their elegant Trigonometric properties, and can also be defined recursively. They are very helpful in Trigonometry, Complex Analysis, and other branches of Algebra.
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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 ...
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Integral of Legendre and Chebyshev polynomials.
I am trying to expand Legendre polynomials into Chebyshev polynomials, shown as:
$$P_{n}(x)=\sum_{k=0}^{n}a_{k}T_{k}(x), $$
where $P_{n}$ is Legendre polynomials and $T_{k}$ is Chebyshev polynomials, ...
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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 ...
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Series Representation of the Glasser function: $\text G(x)\mathop=\limits^\text{def} \int_0^x \sin(t\sin(t))dt\sim2\sqrt{\frac x\pi}$
Here is an uncommon special function called the Glasser function as referenced by Wolfram Mathworld.
which is defined as:
$$\text G(x)\mathop=^\text{def} \int_0^x \sin(t\sin(t))dt\sim2\sqrt{\frac x\pi}...
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First kind Chebyshev polynomial to Monomials
Express First kind Chebyshev polynomial in terms of monomials
First kind Chebyshev polynomial of order n ($T_n$) is defined in terms of cosine function as follow:
1) $T_n(\cos x)=\cos n x$
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relation between first kind Chebyshev poly and second kind Chebyshev poly
How do you prove following relation between Chebyshev poly of first kind and Chebyshev poly of second kind:
$$dT_n(x)/dx=nU_{n-1}(x)$$
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How to use Chebyshev Polynomials to approximate $\sin(x)$ and $\cos(x)$ within the interval $[−π,π]$?
I have approximated $\sin(x)$ and $\cos (x)$ using the Taylor Series (Maclaurin Series) with the following results:
$$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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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
$$ \...
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How to best approximate higher-degree polynomial in space of lower-degree polynomials?
My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)?
Orginially, as the title of the post ...
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Extending a Chebyshev-polynomial determinant identity
The following $n\times n$ determinant identity appears as eq. 19 on Mathworld's entry for the Chebyshev polynomials of the second kind:
$$U_n(x)=\det{A_n(x)}\equiv \begin{vmatrix}2 x& 1 & 0 ...
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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 ...
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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{...
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The relationship between the best uniform approximation and chebyshev interpolant
I am learning Approximation Theory. I know one polynormial of degree at most $n$ is the best uniform approximation of the function $f \in \mathcal C[a,b]$ if and only if there is exist a set of $n+2$ ...
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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=...
user257768
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How to find Chebyshev nodes
I want to use Chebyshev interpolation. But I am a little confused for finding Chebyshev nodes. I use the following figure to illustrate my problem. Consider I have a vector of numbers I depicted as a ...
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