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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37 views

A limit with Chebyshev polynomials

How could I show that this limit: $$\lim_{N\to\infty}\frac{\sum_{p=1}^N T_{4N} \left(u_0(N)\cdot \cos\frac{p\pi}{2N+1}\right)}{N}$$ is equal to 0? In the expression above $T_{4N}$ is the Chebyshev ...
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Show that if $p\in P_n[a,b]$ with leading coefficient $a_n$ then $\|p\|_\infty \geq \frac{|a_n|(b-a)^n}{2^{n-1}}$

Let $T_n\in P_n[-1,1]$ the n-th Chebyshev polynomial. Show that if $p\in P_n[a,b]$ with leading coefficient $a_n$ then $$\|p\|_\infty \geq \frac{|a_n|(b-a)^n}{2^{n-1}}$$ I have been strugguling with ...
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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}\...
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Equation with Chebyshev polynomials

I have a problem that desperately needs solving or some assurance that it is not solvable. The problem is the following: Find an analytic expression for the line in the complex plane that correctly ...
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51 views

Can all Lissajous curves be described by an implicit equation?

I was trying to find the implicit form of the parametric curve $L(x(t),y(t))$ where $x(t)=5\cos\left(3 \pi t +\frac{\pi}{4}\right)$ and $y(t)=3cos\left(6 \pi t -\frac{\pi}{4}\right)$. At the end I ...
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53 views

Show that first order Chebyshev polynomials are in fact polynomials

Given the 1st order Chebyshev polynomials $$ G(x,t) = \sum_{n=0}^{+\infty} T_n(x) t^n = \frac{1-tx}{1-2xt+t^2} $$ I'm wondering how can I show that $T_n(x)$ are polynomials ?
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Show that this integral is a multiple of a Chebyshev polynomial

Consider the function defined by the integral: $$ f_n(x) = -\int_0^\pi \cos(n\theta)\log(|x-\cos\theta|)\,d\theta $$ I want to show that $f_n$ is a multiple of the nth Chebyshev polynomial on the ...
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How to Find the Coefficients For the Chebyshev Expansion of a Function

Find and building a Fourier series of a function $f(x)$ on an arbitrary interval $[a,b]$ is explained here. I know that for Chebyshev series, the expansion is $$f(x) \sim \sum_{i=0}^{N} c_i T_i(x)$$ ...
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How to compute integration with numerical results from Chebyshev spectral method?

I am solving a 2D PDE equation (Poisson's equation) with Chebyshev collocation spectral method. I managed to solved the equations and get the corresponding $\varphi$ values at each Chebyshev nodes. $$\...
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Compositeness testing using Chebyshev polynomials of the second kind

Can you prove or disprove the following claim: Let $U_n(x)$ be Chebyshev polynomial of the second kind and let $a$ be a positive integer greater than one . If $p$ is a prime number such that $p>a+...
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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'})\,...
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low and high k damping for non-fourier methods

I'm working on solving a set of PDEs that describe micro-turbulence in a fusion plasma.There is an article with results that I'm trying to reproduce to make sure my numerical method is working ...
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Chebyshev coefficients upper bounded by sup-norm of function?

Let $T_k(x)$ be the Chebyshev polynomials of the first kind and consider the function $$ f(x) = \sum_{k = 0}^\infty c_k \, T_k(x). $$ Show that $$ |c_k| \leq \|f\|_{[-1,1]}. $$
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Recurrence relation for Chebyshev polynomials of the first kind

I tried to derive recurrence relation for Chebyshev polynomials from their generating function $$\frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}T_{n}(x)t^n.$$ I've differentiated both sides with respect to ...
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Jack d'Aurizio's exercise on Chebyshev polynomials [duplicate]

I am working through Jack D'Aurizio's “Superior Mathematics from an Elementary point of view”, and I found (Lemma 61) the following lemma: $\sum_{k=1}^{n-1}\frac{1}{\sin^2(\pi k/n)}=(n^2-1)/3$. He ...
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Integral involving Chebyshev polynomial

I want to know whether it is possible to show that $\displaystyle\int^{\frac{\pi}{2}}_{0}\cos{t}\cdot U_{2n-1}(\cos{t})\,dt=\frac{\pi}{2}$ for $n\in\mathbb{N}$,where $U_n$ is the Chebyshev polynomial ...
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Eigenvalues of Jacobian of Mandelbulb “triplex” power formula

I'm trying to find a lower bound for the distance estimate of the Mandelbulb fractal, or at least justify why using the scalar-derivative for distance estimation is so effective. The Mandelbulb ...
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Chebyshev interpolating polynomial theorem

Could anyone explain this Chebyshev interpolating polynomial theorem 3.5 ?
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Chebyshev in $n$ dimensions

The Chebyshev monomial integrals (of first and second kind) are $$ I_k = \int_{-1}^1 x^k (1-x^2)^{\mp 1/2} \,dx $$ Is anything known about their $n$-dimensional generalizations $$ I_{k_1,\dots,k_n} = \...
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Chebychev polynomial of $ (-x)$

I know that the Chebyshev polynomials are defined as $T_n(x)=\cos(nx)$ How can I proove the following result: $T_n(-x)= (-1)^n T_n(x)$
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Show that the sign of the sequence is alternating [closed]

I'm solving the following (numerical analysis) problem. Let $f(x)$ be a polynomial of degree $n$ such that there exist exactly $n+1$ distinct points $x_0,x_1\cdots,x_n$ in the interval $[-1,1]$ ...
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Relationship between Poisson's integral formula and the generating function of Chebyshev polynomials

On the disk $\{z:|z|<R\}$, Poisson's integral formula is $$u(r,\theta)=\frac1{2\pi}\int_0^{2\pi}\frac{(R^2-r^2)f(\phi)}{R^2-2Rr\cos(\theta-\phi)+r^2}\,d\phi$$ which solves the Dirichlet problem. ...
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How to go from Fourier to Chebyshev?

I need to convert a Fourier cosine series to Chebyshev. I guess the starting point is to Chebyshev expand $\cos(\pi x)$. Is there a closed-form for the Chebyshev coefficients? I expect the answer is ...
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Understanding the Chebyshev Differential Equation

I'm reading through an article as research for an essay I have been assigned and have found a proof of the Chebyshev Differential Equations being satisfied by Chebyshev polynomials on Wolfram Alpha: ...
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Finding Chebychev coefficients

To find Chebyshev coefficients I need to compute the following polynomial with $T_k(x)$ being the Chebyshev polynomial. We use that $T_k(\cos(z))=\cos(kz)$ $$ 2\int_0^1 T_k(x)\frac{1}{\sqrt{1-x^2}} dx=...
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Integral of Chebyshev polynomial expression $\int_{0}^{1} x^{-1/3}\cos(3\arccos(x))dx$

I am trying to integrate $$\int_{0}^{1} x^{-1/3}\cos(3\arccos(x))dx$$ but I am getting nowhere. The usual $$x=\cos(\theta)$$ substitution and other similar ones I have tried do not seem to work.
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Deriving the recurrence relation for Chebyshev polynomials using law of cosines?

I am trying to derive the recurrence relation in the Chebyshev polynomial using the following recurrence relation: $\cos((n+1)\cos^{-1}x)$ $= x\cos(n\cos^{-1}x) $ - $\sin(n\cos^{-1}x)\sin(\cos^{-1}x)$...
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Bound of Chebyshev polynomials

I'm learning about Chebyshev polynomial (First kind) $T_n$ and I'm confusing with this problem: " Prove that with all $n>5$ we have this inequality: $\vert(\sqrt 3)^{n}\cdot T_n(\dfrac{1}{\sqrt 3}...
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Integral of a Chebyshev polynomial with respect to this special measure (p-adic Plancherel measure for GL2(Q_p))

I am trying to show that the integral $\int_{-2}^2 U_n \left (\frac{x}{2}\right) \frac{p+1}{\pi}\frac{\sqrt{1-\frac{x^2}{4}}}{\left ( \sqrt{p}+\frac{1}{\sqrt{p}} \right )^2 - x^2} dx$ equals $p^{-n/2}$...
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The integrals of the derivatives of the chebyshev polynomials

Now I'm working at an engineering problem involving the integrals of Chebyshev polynomials and its derivatives. For example, \begin{equation} \begin{split} I_1&=\int_{-1}^1T_m(x)T_n(x)\,\mathrm{d}...
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Proof of irrationality of $\arcsin(\frac{1}{4})$

I was working to find a different approach to Niven's theorem from the one in my textbook taking a route via Chebyshev polynomials. It all comes to proving the irrationality of $\arcsin(\frac{1}{4})$ ...
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coefficient by $x^{n-1}$ in Chebyshew polynominals

Calculate coefficient by $x^{n-1}$ in Chebyshew polynominal of the first kind $T_n$, defined as: $$ T_0(x)=1\\ T_1(x)=x\\ T_n(x)=2x\cdot T_{n-1}(x)-T_{n-2}(x) $$
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Chebyshev polynomial statement

So I have been working on this one for a while now and can't seem to even have a clue what to do: Prove the following statement -- Let $T_n(x) = 2^{n−1}P_n(x)$. Then $∀n, m ∈ N, T_n(T_m(x)) = T_{...
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Gauss-Chebyshev quadrature formula

I am currently working to numerically evaluate integral of form $$I=\int_{-1}^{1}\dfrac{f(x)}{\sqrt{1-x^2}}\text{d}x$$ Gauss-Chebyshev formula says $$\int_{-1}^{1}\dfrac{f(x)}{\sqrt{1-x^2}}\text{d}x=\...
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How would one go about computing a Chebyshev-type quadrature problem with general integration limits of [a,b] instead of [-1,1]?

More specifically, a problem of the form $\int_a^b\frac{f(x)}{\sqrt{1-x^2}}dx = \sum_{i=1}^{N}w_if(x_i)$, where $a,b \in [-1,1]$, $w_i$ are the weights, and $x_i$ are the abscissa. A quick search ...
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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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Extremal property of Chebyshev polynomials of second kind

Theorem Let $p(x)=x^n+\dots$ monic polynomial of degree $n$ and $U_n$ n-th Chebyshev polynomial of second kind.Then it holds $$ \|p\|\geq\|\dfrac{1}{2^{n}}U_n\|=\dfrac{1}{2^{n-1}}$$ where $\|f\|=\...
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Extrema of Chebyshev polynomials (of the first kind)

I can hardly find a proof why the extrema of the Chebyshev polynomials are $$ x_k=\cos(\frac{k}{n}\pi), k=1,...n $$ and also why there are $n+1$ of them. The Chebyshev polynomials are here defined as ...
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Chebyshev Polynomials of the first kind

If I write the Chebyshev polynomial of the first kind like this: $T_n(x)=\cos(n\cos^{-1}x)$ for $x$ in $[-1,1]$. It is clear that if $x=1$ then: $T_n(1)=\cos(n\cos^{-1}1)=\cos(0)=1$ for all $n$, ...
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Square roots modulo p

For Chebyshev-polynomials on finite fields $F_p$ with a prime $p$, I have found the following expression: $$ T_n(x)=\frac{(x+\sqrt{x^2-1})^n+(x-\sqrt{x^2-1})^n}{2} mod\space p. $$ But obviously, $\...
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Chebyshev first and second kind recurrence

Hi i have this quantity: $ T_n(x) = exp[i*n(arccos(x)] $ how can i represent the polynomials of first and second kind? I'am trying substituting $n = 0$ and $n = 1$ but i don't know when to stop and ...
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One of Chebyshev's inequalities

How can I prove that this polynomial has at least n+1 zeroes? I have no idea.
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Prove that $\int_1^a \frac{T_n(x) T_n(x/a)}{\sqrt{a^2 - x^2} \sqrt{x^2 - 1^2}} \frac{a}{x} \mathrm{d}x = \frac{\pi}{2}$

In the paper, Representation of a Function by Its Line Integrals, with Some Radiological Applications, A. M. Cormack, Journal of Applied Physics 34, 2722 (1963), an integral identity is expressed ...
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Spectrum self-adjoint operator induced by right shift operator on $\ell^2(\mathbb{N})$.

Let $e_i$ be the standard bais vectors of $\ell^2(\mathbb{N)}$ and let $S$ denote the right shift operator on $\ell^2(\mathbb{N)}$, i.e. $Se_i= e_{i+1}$. Now the operator $T = S + S^*$ is self-adjoint ...
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Chebyshev polynomial generalization for non-integer degrees

I am trying to generalize the Chebyshev polynomials (especially of first kind) for non-integer degree. The properties I would like to keep is $$2 T_m(x) T_n(x) = T_{m+n}(x) + T_{|m-n|}(x)$$ and $$T_m(...
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Computing coefficients of Chebyshev interpolation polynomial in Chebyshev polynomial roots

I'm looking for a method to compute coefficients of Chebyshev interpolation polynomial of function $ f $ $$ p_n(x) = \frac{2}{n+1}( \frac{1}{2}a_0T_0(x) + \sum^n_{i = 1}a_iT_i(x)) $$ where $$ a_i = \...
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Derive the explicit expression of the Chebyshev polynimial: $T_n(x) =n\sum _{k=0}^{n}(-2)^{k}{\frac {(n+k-1)!}{(n-k)!(2k)!}}(1-x)^{k}$.

The explicit expression for the Chebyshev polynomials of the first kind is given as follows. $$ T_n(x) =n\sum _{k=0}^{n}(-2)^{k}{\frac {(n+k-1)!}{(n-k)!(2k)!}}(1-x)^{k}\qquad n>0 $$ However, no ...
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Markov or Chebyshev used Inequality here? stat

Suppose $X =\{−4, −3, −2, −1, 0, 1, 2, 3, 4 \}$ and suppose $E[X] = 0$. Give an upper bound on $P (X = 4)$ and an upper bound on $P (X < −2)$. Solution given as: Let $Y = X + 4$, $E[Y ] = E[X] + ...
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Find all subfix $n$

$a \ge 2$, $a_0=1$, $a_1=a$, $a_2=2a^2-1$, $\frac{a_{n+1}}{a}=a_n+2aa_{n-1}-a_{n-2}, \forall n \ge 2$. Find all subfix $n$, such that there exists such integer $a$, satisfying , among all $a_k \equiv ...
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Classic Complex Numbers - Given $z+\frac 1z=2\cos 3^\circ$, find least integer greater than $z^{2000}+\frac 1{z^{2000}}$

Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ,$ find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}.$ Solution: We have $z=e^{i\theta}$, so $e^{i\theta}+\...

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