Proving expression involving Schwartz function tends to zero

Let $\xi \in \mathbb{R}$ and $f$ belong to the Schwartz space on $\mathbb{R}$. I know, that Schwartz functions are rapidly decreasing, but I am not very familiar with them. My question is, why is it true that $$\lim_{x \to \infty} f(x)(e^{-2\pi i x \xi} - e^{2\pi i x \xi}) = 0$$ I mean we know that for any $k \in \mathbb{N}$ $$\sup_\mathbb{x \in R} |x|^k |f(x)| < \infty$$...Read more

Example of a Schwartz function

If $\psi,\phi \in \mathcal{S}(\mathbb{R}^n)$ then I know that the product $\psi\phi \in \mathcal{S}(\mathbb{R}^n)$ is also in the Schwartzspace.Now I was wondering if $\psi\in \mathcal{S}(\mathbb{R}^n)$ but $\phi \notin \mathcal{S}(\mathbb{R}^n)$ if it is possible for the product $\psi\phi \in \mathcal{S}(\mathbb{R}^n)$ to be in the Schwartz space.I am not sure if this is true, however I think that the multiplication with a non-smooth $\phi$ will always give a non-smooth function back and thus $\psi\phi \notin \mathcal{S}(\mathbb{R}^n)$ How do ...Read more

Is translation continouos in Schwartz space?

It is true that translation is continuous in Schwartz Space {S}($\mathbb{R}$) with its topology?, in other words, I'm trying to prove that if $\phi \in {S}\left(\mathbb{R}\right)$ then the function $\phi(\cdot-y)$ converges to $\phi$ in $S\left(\mathbb{R}\right)$.So I have to prove that for all $\alpha, \beta \in \mathbb{N}$.\begin{equation*} \lim _{ y\rightarrow 0 }{ { \left\| { T }_{ y }\left( \phi \right) -\phi \right\| }_{ \alpha ,\beta } } =0. \end{equation*}This is my attempt.It is known that fourier transformation is continuous in ...Read more

schwartz space - Check if a function is $\mathscr S(\mathbb R)$

How can I check if a generic function is $\mathscr S(\mathbb R)$ ? I mean the Schwartz space.The definition asserts that $f\in \mathscr S(\mathbb R)$ if:$f \in C^\infty (\mathbb R)$$\displaystyle\Vert f \Vert = \sup_{x\in\mathbb R} | x^\alpha D^\beta f(x) |$Is there an easy way to check the assertion 2?...Read more