Taylor Series with Remainder

We repeat the derivation of the preceding section, but this time we treat the error term more carefully.

Again we want to approximate
with an
th-order *polynomial*:

is the ``remainder term'' which we will no longer assume is zero.

Our problem is to find
so as to minimize
over some interval
containing
. There are many
``optimality criteria'' we could choose. The one that falls out
naturally here is called *Padé approximation*. Padé
approximation sets the error value and its first
derivatives to
zero at a single chosen point, which we take to be
. Since all
``degrees of freedom'' in the polynomial coefficients
are
used to set derivatives to zero at one point, the approximation is
termed *maximally flat* at that point. In other words, as
, the
th order polynomial approximation approaches
with an error that is proportional to
.

Padé approximation comes up elsewhere in signal processing. For example, it is the sense in which Butterworth filters are optimal [55]. (Their frequency responses are maximally flat in the center of the pass-band.) Also, Lagrange interpolation filters (which are nonrecursive, while Butterworth filters are recursive) can be shown to maximally flat at dc in the frequency domain [84,37].

Setting in the above polynomial approximation produces

where we have used the fact that the error is to be exactly zero at in Padé approximation.

Differentiating the polynomial approximation and setting gives

where we have used the fact that we also want the

In the same way, we find

for , and the first derivatives of the remainder term are all zero. Solving these relations for the desired constants yields the th-order Taylor series expansion of about the point

as before, but now we better understand the remainder term.

From this derivation, it is clear that the approximation error (remainder
term) is smallest in the vicinity of
. *All degrees of freedom*
in the polynomial coefficients were devoted to minimizing the approximation
error and its derivatives at
. As you might expect, the approximation
error generally worsens as
gets farther away from 0
.

To obtain a more *uniform* approximation over some interval
in
, other kinds of error criteria may be employed. Classically,
this topic has been called ``economization of series,'' or simply
polynomial approximation under different error criteria. In
`Matlab` or
`Octave`, the function
`polyfit(x,y,n)` will find the coefficients of a polynomial
of
degree `n` that fits the data `y` over the points `x` in a
least-squares sense. That is, it minimizes

where .

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