Schwarz Space

Schwarz Space

The Schwarz Space on $\mathbb{R}$ consists of the set of all indefinitely differentiable functions f so that f and all its derivatives $f^{\prime },f^{\prime \prime },...,f^{\ell },...,$ are rapidly decreasing, in the sense that ${\sup }_{x\in \mathbb{R}}|x{|}^k|f^{\ell }(x)|<\infty \text{for every}\ell ,k\ge 0.$ We denote this space by $\mathcal{S}=\mathcal{S}(\mathbb{R})$.

Schwarz Space

Equivalently, the Schwarz Space on $\mathbb{R}$ is a smooth function $f$ that for any $N$, there exist a $C_N$ such that $|f(x)|\le \frac{C_N}{(1+|x|{)}^N}$

Proposition

The Schwarz Space is a vector space over $\mathbb{C}$. Moreover if $f\in \mathcal{S}(\mathbb{R})$, we have $f^{\prime }=\frac{\text{d}f}{\text{d}x}\in \mathcal{S}(\mathbb{R})\text{and}xf\in \mathcal{S}(\mathbb{R}).$ So the Schwartz space is closed under differentiation and multiplication by polynomials.

Remark

An important class of examples in $\mathcal{S}(\mathbb{R})$ are the “bump functions” which vanish outside bounded intervals, also note that although $e^{-|x|}$ decreases rapidly at infinity, it is not differentiable at 0 and therefore does not belong to $\mathcal{S}(\mathbb{R})$.

Gaussian

A simple example of a function in $\mathcal{S}(\mathbb{R})$ is the Gaussian defined by $f(x)=e^{-x^2}.$ In fact, $e^{-a{x}^2}$ belongs to $\mathcal{S}(\mathbb{R})$ whenever $a>0.$