# Exp (-x^2) is Schwartz Test Function

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## Theorem

$\map \exp {-x^2}$ is a Schwartz test function.

## Proof

Let:

- $\dfrac {\d^n}{\d x^n} \map \exp {-x^2} = \map {p_n} x \map \exp {-x^2}$

We have that $\map {p_0} x = 1$ and $\map {p_1} x = -2 x$.

Then:

\(\ds \dfrac {\d^{n + 1} } {\d x^{n + 1} } \map \exp {-x^2}\) | \(=\) | \(\ds \dfrac \d {\d x} \dfrac {\d^n}{\d x^n} \map \exp {-x^2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac \d {\d x} \paren {\map {p_n} x \map \exp {-x^2} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map {p_n'} x \map \exp {-x^2} - 2 x \map {p_n} x \map \exp {-x^2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren{\map {p_n'} x - 2 x \map {p_n} x} \map \exp {-x^2}\) |

where

- $\map {p_{n + 1}} x := \map {p_n'} x - 2 x \map {p_n} x$

Note that each subsequent $\map {p_n} x$ is constructed from derivatives, products and differences of polynomials.

Hence, $\map {p_n} x$ is a polynomial.

Consider the Maclaurin series of $\map \exp {x^2}$ .

\(\ds \map \exp {x^2}\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {x^2}^k} {k!}\) | ||||||||||||

\(\ds \) | \(\ge\) | \(\ds \frac {\paren {x^2}^k} {k!}\) | $k \in \N$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\size x^k} {k!} \size x^k\) | Absolute Value Function is Completely Multiplicative | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \size x^{-k} k!\) | \(\ge\) | \(\ds \size x^k \map \exp{-x^2}\) |

Suppose $\size x \ge 1$.

Then:

\(\ds \size {x^k \map \exp {-x^2} }\) | \(=\) | \(\ds \size x^k \map \exp {-x^2}\) | Absolute Value Function is Completely Multiplicative | |||||||||||

\(\ds \) | \(\le\) | \(\ds \frac {k!} {\size x^k}\) | ||||||||||||

\(\ds \) | \(\le\) | \(\ds k!\) | $\size x \ge 1$ |

Suppose $\size x \le 1$.

We have that $x^k \map \exp {-x^2}$ is a continuous function.

Hence, it is bounded:

- $\forall x \in \closedint {-1} 1 \exists M \in \R : \size {x^k \map \exp {-x^2}} \le M$

Altogether:

- $\sup \size {x^k \map \exp {-x^2}} \le \map \max {M, k!} < \infty$

Thus:

- $\map \exp {-x^2} \in \map \SS \R$.

$\blacksquare$

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): Chapter $\S 6.5$: A glimpse of distribution theory. Fourier transform of (tempered) distributions