There are two notions in the book by Stein and Shakarchi. The Schwartz's space on $\mathbb{R}$ consists of the set of all indefinitely differentiable functions $f$ so that $f$ and all its derivatives $f^{'}, f^{''},\ldots,f^{(\ell)},\ldots,$ are rapidly decreasing in the sense that $$\sup_{x\in \mathbb{R}}|x|^k|f^{(\ell)}(x)|<\infty$$ for every $k,\ell\ge 0.$ A function $f$ defined on $\mathbb{R}$ is said to be of moderate decrease if $f$ is continuous and there exists a constant $A>0$ such that
$$|f(x)|\le \frac{A}{1+x^2}, x\in \mathbb{R}.$$
What is the relationship between the two spaces? Is the Schwartz's space contained in the set of functions of moderate decrease? Can we find a function which is of moderate decrease but not in Schwartz space?
Best Answer
If $f$ is in the Schwarz space then $|f(x)|$ and $x^{2}f(x)$ are bounded. So $(1+x^{2}) f(x)$ is bounded which means $f$ is moderately decreasing.
$f(x)=\frac 1 {1+x^{2}}$ is moderately decreasing but it is not in the Schwarz space becasue $x^{3}f(x)$ is not bounded.