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The th Chebyshev polynomial may be defined by
The first three even-order cases are plotted in Fig.3.29
(We will only need the even orders for making Chebyshev windows.)
. Using the double-angle trig
, it can be verified that
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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA)
, teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/