Flat exponential - Knowlpedia

Core idea

For $c>0$ and a positive integer $q$ , the flat exponential

h(t)=\begin{cases}e^{-c/t^q},&t>0,\\0,&t\le0\end{cases}

is smooth on $\mathbb R$ and flat at zero.

Proof

Every derivative on $t>0$ is a finite sum of terms $C t^{-N}e^{-c/t^q}$ . Each tends to zero as $t\downarrow0$ : put $s=c/t^q$ and use $s^M e^{-s}\to0$ for every fixed $M\ge0$ . One elementary bound follows from the exponential series: $e^{s/2}\ge (s/2)^k/k!$ with an integer $k>M$ .

Extend all these derivatives by zero to the other side. Induction, using difference quotients or the fundamental theorem of calculus, proves that the extensions are successive derivatives of $h$ . This proves smoothness and flatness.

Products and shifts

Translates produce flat endpoints at other locations. Multiplying two such factors yields a smooth function positive on a bounded interval and zero outside it, providing explicit bump functions.

References

Richard Schwartz, Partitions of Unity (Brown lecture notes).