Not long ago, I derived the formula for Chebyshev polynomials
$$T_{n}\left( x\right)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}{n \choose 2k}x^{n-2k}\left( x^2-1\right)^{k}$$
How to extract the coefficients of this polynomial of degree n ?
I tried using Newton's binomial but got a double sum...
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 polynomials of order ##4N##, ##u_0(N)\geq 1## is a number such that ##T_{4N}(u_0)=b##, with...
Hello everyone.
I am studying this article since I am interested in optimization. The article makes use of Clenshaw–Curtis quadrature scheme to discretize the integral part of the cost function to a finite sum using Chebyshev polynomials.
The article differentiates between the case of odd...
Hello everyone. I am trying to construct an optimization problem using Chebyshev pseudospectral method as described in this article. For that, I need to calculate the zeros of the Chebyshev polynomial of any order. In the article is sugested to do it as
tk=cos(πk/N) k=0, ..., N...
Hello everyone. I need to construct in python a function which returns the evaluation of a Chebishev polynomial of order k evaluated in x. I have tested the function chebval form these documents, but it doesn't provide what I look for, since I have tested the third one, 4t^3-3t and
import numpy...
Hey! :o
We are given the polynomial functions $$T_0(x)=1, T_1(x)=x, x \in \mathbb{R} \\ T_{n+1}(x)=2xT_n(x)-T_{n-1}(x), n \in \mathbb{N}, x \in \mathbb{R}$$
(Chebyshev polynomials)
Using induction I have to show that:
the degree of $T_n$ is $n$
$\forall n \in \mathbb{N}$ : $T_n(1)=1$...
This is something Chebyshev polynomial problems. I need to show that:
##\sum_{r=0}^{n}T_{2r}(x)=\frac{1}{2}\big ( 1+\frac{U_{2n+1}(x)}{\sqrt{1-x^2}}\big )##
by using two type of solution :
##T_n(x)=\cos(n \cos^{-1}x)## and ##U_n(x)=\sin(n \cos^{-1}x)## with ##x=\cos\theta##,
I have form the...
So I've been reading about minimax polynomial approximations and have found them to be pretty impressive. However, i am confused on exactly how to determine the constants.
The first step is supposed be solving for the Chebyshev polynomials as an initial guess. I'm reading wikipedia but I'm a...
Question
A Chebyshev polynomial is Tn(x) = cos(arccos^(-1)(x))
My questions are:
1. what are the domain(s) and range(s) of this function?
2. Give equivalent polynomial definitions for Tn(x) when n = 0; 1; 2; 3. That
is: show that the definition for Tn above really is a polynomial...
I was hoping someone could point me in the direction of a suitable extension of Chebyshev polynomials to mutple dimensions?
I find Chebyshev polynomials useful in situations when I need to fit some function in a general way, imposing as little pre-concieved ideas about the form as possible...
Hi,
I fail finding a proof (even in MathWorld, in my Mathematic dictionary or on the Web) for the following property of Chebyshev polynomials:
(T_i o T_j)(x) = (T_j o T_i)(x) = T_ij(x) when x is in ] -inf ; + inf [
Example :
T_2(x) = 2x^2-1
T_3(x) = 4x^3-3x
T_3(T_2(x)) = T_2(T_3(x)) =...