## 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.

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Differentiability of Audio Signals

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Weierstrass Approximation Theorem