Infinite Flatness at Infinity

The Gaussian is infinitely flat at infinity. Equivalently, the Maclaurin expansion (Taylor expansion about $t=0$ ) of

$$f(t) = e^{-\frac{1}{t^2}}$$ (D.3)

is zero for all orders. Thus, even though $f(t)$ is differentiable of all orders at $t=0$ , its series expansion fails to approach the function. This happens because $e^{t^2}$ has an essential singularity at $t=\infty$ (also called a ``non-removable singularity''). One can think of an essential singularity as an infinite number of poles piled up at the same point ($t=\infty$ for $e^{t^2}$ ). Equivalently, $f(t)$ above has an infinite number of zeros at $t=0$ , leading to the problem with Maclaurin series expansion. To prove this, one can show

$$\lim_{t\to 0} \frac{1}{t^k} f(t) = 0$$ (D.4)

for all $k=1,2,\dots\,$ . This follows from the fact that exponential growth or decay is faster than polynomial growth or decay. An exponential can in fact be viewed as an infinite-order polynomial, since

$$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots.$$ (D.5)

We may call $f(t) = e^{-\frac{1}{t^2}}$ infinitely flat at $t=0$ in the ``Padé sense'':

• Padé approximation is maximally flat approximation, and seeks to use all $N$ degrees of freedom in the approximation to match the $N$ leading terms of the Taylor series expansion.
• Butterworth filters (IIR) are maximally flat at dc [263].
• Lagrange interpolation (FIR) is maximally flat at dc [266].
• Thiran allpass interpolation has maximally flat group delay at dc [266].

Another interesting mathematical property of essential singularities is that near an essential singular point $z_0\in\mathbb{C}$ the inequality

$$\left\vert f(z)-c\right\vert<\epsilon$$ (D.6)

is satisfied at some point $z\neq z_0$ in every neighborhood of $z_0$ , however small. In other words, $f(z)$ comes arbitrarily close to every possible value in any neighborhood about an essential singular point. This was first proved by Weierstrass [42, p. 270].