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Chebyshev Polynomials
Figure 3.34:

The
th Chebyshev polynomial may be defined by

(4.46) 
The first three evenorder 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 doubleangle trig
formula
, it can be verified that

(4.47) 
for
.
The following properties of the Chebyshev polynomials are well known:

is an
thorder 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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