Chebyshev Polynomial Properties

• $T_n(x)$ is an $n$ th-order polynomial in $x$
• $T_n(x)$ is an even function when $n$ is an even integer, and odd when $n$ is odd
• $T_n(x)$ has $n$ zeros in the open interval $(-1,1)$ , and $n+1$ extrema in the closed interval $[-1,1]$
• $T_n(x)>1$ for $x>1$

$$T_n(x) = \cos[n\cos^{-1}(x)], \quad \vert x\vert\le1$$