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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Question about chebyshev polynomial
Chebyshev polynomials are defined as such:
$$T_n(x)=\cos(n\arccos(x))$$
I'm asked to show that $\deg(T_j(x))=j$ and that $T_0,T_1,T_2,\ldots,T_n$ are an orthogonal basis of $\Bbb R_n[x]$.
I think I ...
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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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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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Coefficients of Chebychev Polynomials
Is there a known formula for the coefficient of x^k in the nth chebychev polynomial of the first kind?
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Prove $T_n(x)$ of Chebyshev Polynomial given the recurrence relation
Using the recursion formula for Chebyshev polynomials, show that $T_n(x)$ can be written as
$$T_n(x)=2^{n-1}(x-x_1)(x-x_2)...(x-x_n)$$
where $x_i$ are the $n$ roots of $T_n$
The recurrence relation:...
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Calculation of Chebyshev coefficients
The Chebyshev polynomials can be defined recursively as:
$T_0(x)=1$;
$T_1(x)=x$;
$T_{n+1}(x)=2xT_n(x) + T_{n-1}(x)$
The coefficients of these polynomails for a function, $\space f(x)$, under ...
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Trigonometric Identities for $\sin nx$ and $\cos nx $
These are generalizations of simple trigonometric identities for $\sin 2x$ and $\cos 2x$, but in general how can we prove them?
$$\sin nx =\sum_{k=1}^{\left\lceil\frac{n}{2}\right\rceil}(-1)^{k-1}\...
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How to best approximate higher-degree polynomial in space of lower-degree polynomials?
My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)?
Orginially, as the title of the post ...
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Integrating Chebyshev polynomial of the first kind
I'm trying to evaluate the integral of the Chebyshev polynomials of the first kind on the interval $-1 \leq x \leq 1 $ .
My idea is to use the closed form
$$T_n(x) = \frac{z_1^n + z_2^{-n} }{2}$$
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