NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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We have a function 
and we want to approximate it using an
th-order
polynomial:
where 
, which is obviously the approximation error, is
called the ''remainder term.'' We may assume 
and
are
real, but the following derivation generalizes unchanged to
the complex case.
Our problem is to find fixed constants 
so as to obtain the best approximation possible. Let's
proceed optimistically as though the approximation will be perfect,
and assume 
for all (
), given the right values of 
. Then
at 
we
must have
That's one constant down and 
to go!
Now let's look at the first derivative of with
respect to ,
again assuming that :
Evaluating this at gives
In the same way, we find
where 
denotes the th
derivative of
with respect to,
evaluated at .
Solving the above relations for the desired constants yields
Thus, defining 
(as it always is), we have derived the following
polynomial approximation:
This is the th-order Taylor series expansion of about
the point. Its
derivation was quite simple. The hard part is showing that the
approximation error (remainder term ) is small over a wide interval of values.
Another ''math job'' is to determine the conditions under which the
approximation error approaches zero for all as the
order goes to
infinity. The main point to note here is that the form of the
Taylor series expansion itself is simple to derive.
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