Let $\mathcal{P}$ be distribution of some random variable $\xi$.
I would like to know, is given function $\varphi(x)$ characteristic or not: $$\varphi(t)=\int_{-\infty}^{\infty} e^{-(tx)^2/2}\ \mathcal{P}(dx).$$
$\varphi(0)=1$, so i assume it is. I see that given function share some similarities with standart normal CF, but i can't get anything useful from that.
I presume that $\xi$ is one-dimensional random variavle.
Let us first consider the case when $\mathcal P$ is discrete distribution with p.m.f. $p_k=\mathcal P(\{a_k\})$, $\sum_k p_k=1$. Then $$ \varphi(t) = \mathbb Ee^{-t^2\xi^2/2}=\sum_{k} p_ke^{-t^2a_k^2/2}. $$ This is characteristic function of a product $\xi\eta$ of two independent r.v.'s, where the distribution of $\eta$ is standard normal.
For the general case, let $\mathcal P$ is the distribution of random variable $\xi$. Let the distribution of $\eta$ is standard normal and $\eta$ and $\xi$ are independent. Then $$ \varphi_{\xi\eta}(t)=\mathbb Ee^{it\xi\eta}=\mathbb E\underbrace{\left(\mathbb E(e^{it\xi\eta}\mid \xi)\right)}_{\varphi_{\eta}(t\xi)}=\mathbb E\left(e^{-t^2\xi^2/2}\right)= \int_{-\infty}^\infty e^{-t^2 x^2/2}\,\mathcal P(dx) $$ Finally, $\varphi(t)$ is a characteristic function of a product of two independent random values: r.v. with distribution $\mathcal P$ and with standard normal distribution.