2

A function $u:\mathbb{R}^n\rightarrow \mathbb{C}$ belongs to the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ if \begin{equation*} \sup_{x\in \mathbb{R}^n} |x^{\alpha} \partial^{\beta} u |<\infty \end{equation*} for all multi-indices $\alpha,\beta\in \mathbb{Z}^{+}$. Typical examples are the Gaussian $x\mapsto e^{-a |x|^2}$, $a>0$, and smooth functions with compact support traditionally denoted by $C^{\infty}_{c}(\mathbb{R}^n)$.

I wonder if there are more known explicit examples of functions in $\mathcal{S}(\mathbb{R}^n)$.

In fact, I wonder if there are explicit examples for functions in $C^{\infty}_{c}(\mathbb{R}^n)$ other than the mollifiers generated by the smooth function $t\mapsto e^{-\frac{1}{t}}$.

Of course, if we mollify any $L^{p}$ function, $p\geq 1$, we obtain an approximating sequence of functions in $C^{\infty}_{c}(\mathbb{R}^n)$. But could any one give more explicit examples ?

Medo
  • 3,070
  • 12
  • 28

1 Answers1

2

There are many examples, like $e^{-p(x)}$, where $p$ is a polynomial of even degree and positive leading coefficient, or $e^{-(\log(1+x^2))^n}$. Other explicit examples can be constructed as products of a previous example with a $C^\infty$ function with bounded derivatives of all orders, like $\sin x$ or $\cos x$.

As for functions of compact support, you can construct mollifiers starting with $t\mapsto\phi(t)$ such that $\phi^{(n)}(0)=0$ for all $n\in\Bbb N$. Also, the product of a function in $C_c^\infty(\Bbb R^n)$ and a smooth function is again in $C_c^\infty(\Bbb R^n)$.

Julián Aguirre
  • 76,354
  • Aguirre. Thanks a lot. Of course, your $p$ can be any smooth positive function whose derivatives do not grow faster than polynomials. But again these are all based on the Gaussian $e^{-|.|^2}$. – Medo Mar 16 '18 at 18:49
  • 1
    Since Schwartz functions decrease at $\infty$ faster that any power $|x|^–-a}$, $a>0$, it is in the nature of things that they have to behave at $\infty$ like $e^{-\text{something}}$, where "something" grows faster that $\log|x|$. – Julián Aguirre Mar 17 '18 at 17:13