Showing $\cos(t^2)$ is not a Characteristic Function

Usually when we try to show a function is not a characteristic function, we would prove it is not uniformly continuous. I am wondering if there is any other way to show $\cos(t^2)$ is not a characteristic function. $%fooling edit check$

Solution 1:

Hint: Put $\varphi(t)=\cos(t^2)$. If $X$ is a random variable with this "characteristic function", we'd have $$\mathbb{E}(X^2)=-\varphi^{\prime\prime}(t)|_{t=0}=0.$$

Solution 2:

Introduce the notation $c(t)=\cos(t^2)$. By Bochner's theorem, every determinant $D(t)$ should be nonnegative, where $$ D(t)=\text{det}\begin{pmatrix}c(0) & c(t) & c(2t)\\ c(-t) & c(0) & c(t)\\ c(-2t) & c(-t) & c(0)\end{pmatrix}. $$ Since $c(t)=c(-t)$ for every $t$, $D(t)=(1-c(2t))E(t)$ with $E(t)=1-2\cos(t^2)^2+\cos(4t^2)$. For $t=1$, $E(1)=1-2\cos(1)^2+\cos(4)=-0.24...$ hence $D(1)<0$.

Alternatively, $E(t)=-6t^4+O(t^8)$ when $t\to0$. Or, $E(t)=-2\sin^2(t^2)(1+2\cos(2t^2))$.