Infinite Flatness at Infinity

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

$\displaystyle f(t) = e^{-\frac{1}{t^2}}$

is zero for all orders. Thus, even though is differentiable of all orders at , 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, above has an infinite number of zeros at , leading to the problem with Maclaurin series expansion. To prove this, one can show

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

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

$\displaystyle e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots.$

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

Padé approximation is maximally flat approximation, and seeks to use all degrees of freedom in the approximation to match the leading terms of the Taylor series expansion.
Butterworth filters (IIR) are maximally flat at dc [247].
Lagrange interpolation (FIR) is maximally flat at dc [251].
Thiran allpass interpolation has maximally flat group delay at dc [251].

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

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

is satisfied at some point $z\neq z_0$ in every neighborhood of , however small. In other words, comes arbitrarily close to every possible value in any neighborhood about an essential singular point. This was first proved by Weierstrass [41, p. 270].

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
Gaussian Function Properties
Infinite Flatness at Infinity