Points of Infinite Flatness

Consider the inverted Gaussian pulse,^E.1

$$f(x) \isdef e^{-1/x^2}, % \frac{1}{x^2}},$$

i.e., $f(1/x)$ is the well known Gaussian ``bell curve'' $e^{-x^2}$. Clearly, derivatives of all orders exist for all $x$. However, it is readily verified that all derivatives at $x=0$ are zero. (It is easier to verify that all derivatives of the bell curve are zero at $x=\infty$.) Therefore, every finite-order Maclaurin series expansion of $f(x)$ 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, $f(x)$ is infinitely flat at $x=0$.

Another property of $f(x) \isdef e^{-\frac{1}{x^2}}$ is that it has an infinite number of ``zeros'' at $x=0$. The fact that a function $g(x)$ has an infinite number of zeros at $x=x_0$ can be verified by showing

$$\lim_{x\to x_0} \frac{1}{(x-x_0)^k} g(x) = 0$$

for all $k=1,2,\dots\,$. For $f(x)$, the existence of an infinite number of zeros at $x=0$ is easily shown by looking at the zeros of $f(1/x)$ at $x=\infty$, i.e.,

$$\lim_{x\to\infty} x^k f(1/x) = \lim_{x\to\infty} x^k e^{-x^2} = 0$$

for any integer $k$. Thus, the faster-than-exponential decay of a Gaussian bell curve cannot be outpaced by the factor $x^k$, for any finite $k$. 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 $e^x$ is $1 + x + x^2/2 + x^3/3! + \dots$--an ``infinite-order'' polynomial.)

The reciprocal of a function containing an infinite-order zero at $x=x_0$ has what is called an essential singularity at $x=x_0$ [13, p. 157], also called a non-removable singularity. Thus, $1/f(x) = e^{\frac{1}{x^2}}$ has an essential singularity at $x=0$, and $e^{x^2}$ has one at $x=\infty$.

An amazing result from the theory of complex variables [13, p. 270] near an essential singular point $z_0\in{\bf C}$ (i.e., $z_0$ may be a complex number), is that the inequality

$$\left\vert f(z)-c\right\vert<\epsilon$$

is satisfied at some point $z\neq z_0$ in every neighborhood of $z_0$, 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 [13].

In summary, a Taylor series expansion about the point $x=x_0$ will always yield a constant approximation when the function being approximated is infinitely flat at $x_0$. For this reason, polynomial approximations are often applied over a restricted range of $x$, with constraints added to provide transitions from one interval to the next. This leads to the general subject of splines [79]. 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.