Questions tagged [chebyshev-polynomials]

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

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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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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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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 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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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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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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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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Chebyshev coefficients from a polynomial

Is there an efficient algorithm for finding the coefficients in a Chebyshev basis of a polynomial? That is, given the set of $a_k$ such that: $p_n(x) = \sum_{k=0}^N a_k x^k $ Find the set of $c_k$ ...
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What is the group-like structure on $x^2+y^2+z^2-2xyz=1$?

(Background: this is inspired by Chebyshev polynomials and expanding a function as a Chebyshev series.) Solving for $ z $ gives $$ z=xy \pm \sqrt{(1-x^2)(1-y^2)}, $$ where $-1\leq x,y \leq 1$. Now ...
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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=...
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Chebyshev polynomial. Show : $ T_n(x) = \frac{1}{2} ((x+\sqrt{x^2-1})^n+(x-\sqrt{x^2-1})^n)$

Consider $T_n(x) = \cos ( n \cdot \arccos(x)) $ on $ I = [-1,1]$. Show: a: $T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x) $ b : The $T_n$ are orthogonal with $(f,g) = \int_{-1}^{1} f(x)g(x)\frac{1}{\sqrt{1-x^...
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integration of chebyshev polynomials of first kind with an exponential funcion

In my class, my tutor raised a question of the following integral: $\int T_n(x)*\exp(a*x)dx,$ where $T_n(x)$ is an n power of Chebyshev polynomials of first kind and a is a constant. The hint he gave ...
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Inner Product of Chebyshev polynomials of the second kind with $x$ as weighting

I have tried to solve the integral $${\int_0^1 U_n (x) U_m (x)x dx },$$ where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind. Solving the integration and checking the result, I ...
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On the extrema of Chebyshev polynomials of the second kind

I wish to prove that the magnitude of extreme values of $U_n(x)$, the Chebyshev polynomial of the second kind, is monotonically increasing on $[-1,1]$. By symmetry it suffices to prove it over $[0,1]$....
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Cubic Approximation to $e^x$ using Chebyshev Polynomial

Was trying to solve this: $C_r=\frac{2}{\pi}\int_{-1}^1\frac{e^xT_r(x)}{\sqrt{1-x^2}}dx$ where $r=0,1,2,3$ $T_r(x) =cosr[{cos}^{-1}x]$ While solving, I equated $x=cos\theta$ Therefore ...
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Transform complex trigonometric expression with $\arccos$

In the proof that the poles of a Chebyshev filter lie on an ellipse, there is the following transformation, for the $s$ values correspondant to the poles. From (1) $$s_{pm} = j \cos \left[ \...