Chebyshev polynomial

$T_n(\cos \theta )=\cos (n\theta )$. These polynomials are orthogonal for weight $(1-x^2{)}^{-1/2}$ and support spectrally accurate approximation.

The Chebyshev polynomial of the second kind is defined byIt is orthogonal on $[-1,1]$ with weight $\sqrt{1-x^2}$ and has roots $\cos (j\pi /(n+1))$ for $1\le j\le n$.

The number $2\cos (\pi /n)$ is an algebraic integer. If it is rational it must be an integer, which shows that $\cos (\pi /n)$ is rational only for $n=1,2,3$.

Every root of $T_n^{(k)}$ lies in $(-1,1)$ and its leading coefficient is positive. Hencefor $x\ge 1$ and $0\le k\le n$.

Among monic real polynomials of degree $n$, $2^{1-n}{T}_n$ uniquely minimizes the supremum norm on $[-1,1]$, attaining the value $2^{1-n}$.

If $f$ has degree at most $n$ and $|f|<1$ on $[-1,1]$, alternation at the extrema of $T_n$ implies $|f(t)|<|T_n(t)|$ outside that interval.

The equation $(1-z^2)w^{\prime \prime }-z{w}^{\prime }+n^{2}w=0$ is obtained from the harmonic equation $w_{xx}+n^{2}w=0$ by $z=\cos x$.

The Chebyshev equation has self-adjoint formFor $U_n=T_n^{\prime }$, differentiation givesThe respective orthogonality weights are $(1-x^2{)}^{-1/2}$ and $(1-x^2{)}^{1/2}$.