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

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

for all . For , the existence of an infinite number of zeros at is easily shown by looking at the zeros of at ,

for any integer . Thus, the faster-than-exponential decay of a Gaussian bell curve cannot be outpaced by the factor , for any finite . In other words, exponential growth or decay is faster than polynomial growth or decay. (As mentioned in §3.10, the Taylor series expansion of the exponential function is --an ``infinite-order'' polynomial.)

The reciprocal of a function containing an infinite-order zero at
has what is called an *essential singularity* at
[16, 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
[16, p. 270]
is that near an essential singular point
(*i.e.*,
may be
a complex number), the inequality

is satisfied at some point in

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*
[84]. 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.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University