#### Chebyshev Polynomials

The th Chebyshev polynomial may be defined by

 (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, and . Using the double-angle trig formula , it can be verified that

 (4.47)

for . The following properties of the Chebyshev polynomials are well known:
• is an th-order polynomial in .
• is an even function when is an even integer, and odd when is odd.
• has zeros in the open interval , and extrema in the closed interval .
• for .

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