# 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.

259 questions
Filter by
Sorted by
Tagged with
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 ...
15 views

### 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 ...
50 views

34 views

61 views

32 views

### 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 ...
60 views

### One of Chebyshev's inequalities

How can I prove that this polynomial has at least n+1 zeroes? I have no idea.
182 views

### 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 ...
56 views

### 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 ...
59 views

100 views

### 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 ...
### 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}+\...