# Taylor Series Expansions

A *Taylor series expansion* of a continuous function is a
*polynomial approximation* of . This appendix derives
the Taylor series approximation informally, then introduces the
remainder term and a formal statement of Taylor's theorem. Finally, a
basic result on the completeness of polynomial approximation is
stated.

## Informal Derivation of Taylor Series

We have a function and we want to approximate it using an
th-order *polynomial*:

*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

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:

## 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*:

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 [53]. (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 [82,36].

Setting in the above polynomial approximation produces

Differentiating the polynomial approximation and setting gives

*slope*of the error to be exactly zero at .

In the same way, we find

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

## Formal Statement of Taylor's Theorem

Let be continuous on a real interval containing (and ), and let exist at and be continuous for all . Then we have the following Taylor series expansion:

where
is called the *remainder term*. Then Taylor's
theorem [63, pp. 95-96] provides that there exists some
between and such that

When , the Taylor series reduces to what is called a *Maclaurin
series* [56, p. 96].

## Weierstrass Approximation Theorem

The Weierstrass approximation theorem assures us that polynomial approximation can get arbitrarily close to any continuous function as the polynomial order is increased.

Let be continuous on a real interval . Then for any , there exists an th-order polynomial , where depends on , such that

For a proof, see, *e.g.*, [63, pp. 146-148].

Thus, any continuous function can be approximated arbitrarily well by
means of a polynomial. This does not necessarily mean that a Taylor
series expansion can be used to find such a polynomial since, in
particular, the function must be differentiable of all orders
throughout . Furthermore, there can be points, even in infinitely
differentiable functions, about which a Taylor expansion will not
yield a good approximation, as illustrated in the next section. The
main point here is that, thanks to the Weierstrass approximation
theorem, we know that good polynomial approximations *exist* for
any continuous function.

## Points of Infinite Flatness

Consider the inverted Gaussian pulse,^{E.1}

*i.e.*, is the well known Gaussian ``bell curve'' . Clearly, derivatives of all orders exist for all . However, it is readily verified that

*all*derivatives at are zero. (It is easier to verify that all derivatives of the bell curve are zero at .) Therefore, every finite-order Maclaurin series expansion of is the zero function, and the Weierstrass approximation theorem cannot be fulfilled by this series.

As mentioned in §E.2, a measure of ``flatness'' is the number of leading zero terms in a function's Taylor expansion (not counting the first (constant) term). Thus, by this measure, the bell curve is ``infinitely flat'' at infinity, or, equivalently, is infinitely flat at .

Another property of is that it has an infinite number of ``zeros'' at . The fact that a function has an infinite number of zeros at can be verified by showing

*i.e.*,

The reciprocal of a function containing an infinite-order zero at
has what is called an *essential singularity* at
[15, p. 157], also called a
*non-removable
singularity*. Thus,
has an essential
singularity at , and has one at .

An amazing result from the theory of complex variables
[15, p. 270]
is that near an essential singular point
(*i.e.*, may be
a complex number), the inequality

*every*neighborhood of , however small! In other words, the function comes arbitrarily close to every possible value in any neighborhood about an essential singular point. This result, too, is due to Weierstrass [15].

In summary, a Taylor series expansion about the point will
always yield a constant approximation when the function being
approximated is infinitely flat at . For this reason, polynomial
approximations are often applied over a restricted range of , with
constraints added to provide transitions from one interval to the
next. This leads to the general subject of *splines*
[81]. In particular, *cubic spline* approximations
are composed of successive segments which are each third-order polynomials. In each segment,
four degrees of freedom are available (the four polynomial
coefficients). Two of these are usually devoted to matching the
amplitude and slope of the polynomial to one side, while the other two
are used to maximize some measure of fit across the segment. The
points at which adjacent polynomial segments connect are called
``knots'', and finding optimal knot locations is usually a relatively
expensive, iterative computation.

## Differentiability of Audio Signals

As mentioned in §3.6, every audio signal can be regarded as infinitely differentiable due to the finite bandwidth of human hearing. That is, given any audio signal , its Fourier transform is given by

*i.e.*, . Clearly, all audio signals in practice are absolutely integrable. The inverse Fourier transform is then given by

Since the length of the integral is finite, there is no possibility that it can ``blow up'' due to the weighting by in the frequency domain introduced by differentiation in the time domain.

A basic Fourier property of signals and their spectra is that
*a signal cannot be both time limited and frequency limited.*
Therefore, by conceptually ``lowpass filtering'' every audio signal to
reject all frequencies above kHz, we implicitly make every audio
signal last forever! Another way of saying this is that the ``ideal
lowpass filter `rings' forever''. Such fine points do not concern us
in practice, but they are important for fully understanding the
underlying theory. Since, in reality, signals can be said to have a
true beginning and end, we must admit that all signals we really work
with in practice have infinite-bandwidth. That is, when a signal is
turned on or off, there is a spectral event extending all the way to
infinite frequency (while ``rolling off'' with frequency and having a
finite total energy).^{E.2}

In summary, audio signals are perceptually equivalent to bandlimited signals, and bandlimited signals are infinitely smooth in the sense that derivatives of all orders exist at all points time .

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Logarithms and Decibels

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