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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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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692 views

Explanation of numpy's Chebyshev curve fit

I'm writing a mini-library in C++ to find a 4th order Chebyshev polynomial (of the first kind) curve fit on set of x/y points varying in size (between 5-36 sets of points). I have found a pretty good ...
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55 views

In Chebyshev general solution, why the constants are set to ${0,1}$?

Chebyshev DE: $$(1-x^2)y''-xy'+n^2y=0$$ The general solution for the DE is: $$y(t)=C\cos(nt+\alpha)$$ I found that books sets the constants to be: ${C=1, \alpha=0}$ Why we set these values ? ...
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Prove: $ \|\hat{p}_n\|_\infty \leq \|p\|_\infty$ for all $p \in \mathbb{P}^a_n$

Let $a>1$ and $\mathbb{P}^a_n = \{p \in \mathbb{P}_n: p(a)=1 \}$. Define $\hat{p}_n \in \mathbb{P}^a_n$ by $ \hat{p}_n = \frac{T_n(x)}{T_n(a)}$, where $T_n$ is the Chebyshev polynomial of ...
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Proving a bound for the leading coefficient of a polynomial.

Show that every real polynomial $x\in C[a,b]$ of degree $n\ge 1$ with leading term $\beta_n t^n$ satisfies $$||x||\ge |\beta_n|\frac{(b-a)^n}{2^{2n-1}}.$$ I am having difficulty proving this. Here on ...
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1k views

Any differences between Lagrange polynomial on Chebyshev points and Chebyshev polynomial?

I have to find an approximate continuous function that passes through a number of points. Many have said that the best (for my specific problem) is to use Chebyshev polynomial decomposition. I have ...
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75 views

Translation and multiplicative identities to Chebyshev polynomials

I am interested on finding translation and multiplicative identities for Chebyshev polynomials in the way they exist for Bernoulli ones: $$T_n (x+y) = ?$$ $$T_n (ax) = ?$$ Thanks
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122 views

Recurrence relation of Chebyshev nodes

Doing a few practice questions, I came across a proof based question. How would one go around to solve this? "Prove the following recurrence relation:" $\frac{d}{dx}T_{n+1}(x)=(n+1)T_n(x)+\frac{n+1}...
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137 views

What happens to Chebyshev polynomials integration when n=1

The integration of Chebyshev polynomials of the first kind has the following value, $$\int T_{n}(x) \, dx = \frac{1}{2} \, \left( \frac{T_{n+1}(x)}{n+1} - \frac{T_{n-1}(x)}{n-1} \right)$$ what happens ...
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550 views

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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121 views

Is this polynomial irreducible over the rationals?

Prove (or disprove): Define $T_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$. If $p$ is an odd prime, then $\sqrt{\frac{T_p(x)-1}{x-1}}$ is an irreducible polynomial over the ...
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118 views

Combinations of Chebyshev polynomials and sin functions

By chance, I see this formula $\int_0^1 T_{2n+1}(x)\sin(ax) { dx \over \sqrt{1-x^2}}=(-1)^n\frac{\pi}{2}J_{2n+1}(a)$ but what is the closed form if we have $\int_0^1 T_{2n}(x)\sin(ax) { dx \over \...
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114 views

dividing a chebyshev polynomial by another polynomial

If I took a Chebyshev polynomial, is it possible to divide it completely by something that isn't a chebyshev polynomial? edit - the question was answered but people were not sure about what I was ...
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355 views

How to improve stability of numerical solutions to partial differential equations

This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction $t$ and one spatial direction $z$ and the numerical ...
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66 views

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 the orthogonality relation of Chebyshev polynomials of the first kind

The Chebyshev polynomials of the first kind are obtained from the recurrence relation $$\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x)~.\end{aligned}$$ Prove ...
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34 views

Chebyshev expansion of $f(x)=\frac{1}{1+(x-s)^2}$

The Chebyshev polynomials of the first kind are obtained from the recurrence relation $$\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x)~.\end{aligned}$$ I ...
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Recurrent formula for Chebyshev polynomial on general range

I'm trying to compute Chebyshev polynomial coefficients on [a, b] and tried to change default recurrent formula $$T_{n+1} = 2xT_n - T_{n-1}$$ to $$T_{n+1} = (2x-(b+a))T_n - \frac{(b-a)}{4}^{2} T_{n-1} ...
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32 views

Minmax approximation using Chebyshev polynomial

Find minmax approximation to f(x)=|x| in $P_3$ on [-1,1] .What is the minmax error I do not know how to do that ,i got hint that i have to use Chebyshev polynomial for approximating |x|
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Bound for coefficients of chebyshev interpolation in standard basis.

Assume we have for some (sufficiently) smooth function $h:[-1,1] \rightarrow \mathbb{C}$ the Chebyshev interpolation of degree $n \in \mathbb{N}$ \begin{equation} P_n(x) = \sum_{i=0}^n c_iT_i(x). \end{...
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29 views

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_{...
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Rescaled Chebyshev Polynomials have the smallest maximum

Let $P^n$ be the set of polynomials $P^n \equiv \{p\, | \deg p = n, \, p(0) = 1\} $. Let $[m, M]$ be an interval where $0 \lt m \lt M$. I want to find $\arg \min_{p \in P^n}( \max_{x\in[m, M]}p(x))$. ...
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27 views

Use the formula for Chebyshev polynomials $T_j(x)$ to determine the coefficients $a_0,\dots,a_n$ in terms of $A_0,\dots,A_{n+1}$

I'm not sure how to proceed on part b. How do I find coefficients without an integer, like say 5, given for an upper bound? I'm also not sure how to express $a_0,...,a_n$ in terms of $A_0,...,A_{n+1}$....
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43 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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24 views

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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42 views

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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24 views

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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35 views

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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8 views

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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47 views

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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23 views

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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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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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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Chebyshev approximation - from any equation

So let's say I have a function: $f(x) = {e^x}$ How do I use Chebyshev polynomials up to order 4, to find the corresponding coefficients? how do I make an approximation equation using these 4 ...
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44 views

Give the upper limit of the difference between truncated power series and economized one (economized using Chebyshev polynomials).

The question is as follow: (a) The first three Chebyshev polynomials are: $$T_0=1$$ $$T_1 = x$$ $$T_2 = 2x^2-1$$ $$T_3 = 4x^3-3x$$ $$T_4 = 8x^4-8x^2+1$$ i) Economize the truncated power series: $$p(...
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Is it possible to solve tridiagonal Toeplitz matrix whose center element is different, using Chebyshev polynomial of the second kind?

I have a tridiagonal Toeplitz matrix whose first diagonal below main diagonal, and the first diagonal above the main diagonal have elements equal to $-1$ and the main diagonal elements are same ...
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43 views

How to find norm $||U_n||$ of Chebyshev polynomials of the second kind?

how to find norm $||U_n||$ and the values $U_n(\pm)$ of the Chebyshev polynomials of the second kind?
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48 views

General Chebyshev approximation

I am having trouble understanding it, first of all, what is x? Are x's coefficients of this polynomial we are looking for? This would mean that the polynomial is of degree $n-1$ because it has n ...
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36 views

What are $\cos(\omega_k), -\sin(\omega_k)$ in Chebyshev filter design in matrix form?

What are $\cos(\omega_k), -\sin(\omega_k)$ in Chebyshev filter design in matrix form? The Chebyshev filter design problem "via SOCP" (https://en.wikipedia.org/wiki/Second-order_cone_programming) is ...
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147 views

Chebyshev approximation for large interval

In the context of neural networks and cryptography, I would like to approximate some activation functions. However, I need to approximate them into polynomial forms for my purpose. It seems that ...
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68 views

Chebyshev's polynomial approximation

Please help me with this question. I want to run a system that will give me answer using MATLAB but I am struggling. Write a function Cheb(n,x) for evaluating $T_n(x)$. Use the recursive formula ...
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373 views

Lagrangian interpolation at Chebyshev points - estimate on coefficients in monomic basis

First, let us fix some Notation: Let $n\in\mathbb{N}$ and $x_i=\cos(\tfrac{(i+1/2)\pi}{(n+1)})$, $i=0,\dots,n$, be the Chebyshev points. Let \begin{align}L_i(x)={\displaystyle\prod_{\substack{0\leq j\...
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80 views

6th Degree Polynomial and Chebyshev minmax

I have to find the 6th degree polynomial for the function $f(x)=xe^x$. After which the use of the Chebyshev min-max approach I have to use the list grade polynomial approach with respect to the fault ...
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58 views

Does $\sum\limits_{i=1}^{n} f(\lambda_i) = \text{tr}(f(A))$ hold for any function $f$?

We know that given a $n$-by-$n$ matrix $A$, and its eigen values $\{\lambda_i\}_1^n$ its trace $\text{tr}(A)$. then the following holds: $$ \sum\limits_{i=1}^{n} \lambda_i = \text{tr}(A) $$ then ...
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114 views

Relationship between Chebyshev Polynomials of first and second types

How do I prove this relation between the first and second kinds of Chebyshev Polynomials: Given $U_n(x) = \frac{\sin ((n+1) arccos(x))}{\sin (arccos(x))} $ Show that $T_{n+1}(x) = (n+1)U_n(x)$
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71 views

Chebyshev polynomial multiplication with size reduction

I have two one dimensional functions which are represented by some polynomial expansions (let's assume Chebyshev polynomials) $f(x)=\sum_{k=1}^{N} f_k T_k(x)$ and $g(x)=\sum_{k=1}^{N} g_k T_k(x)$. I ...
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51 views

Chebyshev's, Gauss' and Vysochanski˘i-Petunin Inequalities for uniform and normal distribution

so I am being asked to compare the bounds obtained by these three inequalities mentioned in the title for a r.v. following a uniform distribution [0,1] and a r.v. following N(u,1) I am struggling ...