Chebyshev polynomials Definition and 10 Threads

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...

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...

i find that chebyshev polynomials are quite useful in numerical computations is there any good references?

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)) =...