Chebyshev Polynomials

Figure 3.34:

The $n$ th Chebyshev polynomial may be defined by

$$T_n(x) = \left\{\begin{array}{ll} \cos[n\cos^{-1}(x)], & \vert x\vert\le1 \\ [5pt] \cosh[n\cosh^{-1}(x)], & \vert x\vert>1 \\ \end{array} \right..$$ (4.46)

The first three even-order cases are plotted in Fig.3.35. (We will only need the even orders for making Chebyshev windows, as only they are symmetric about time 0.) Clearly, $T_0(x)=1$ and $T_1(x)=x$ . Using the double-angle trig formula $\cos(2\theta)=2\cos^2(\theta)-1$ , it can be verified that

$$T_n(x) = 2x T_{n-1}(x) - T_{n-2}(x)$$ (4.47)

for $n\ge 2$ . The following properties of the Chebyshev polynomials are well known:

• $T_n(x)$ 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 $x>1$ .