Convolution of a probability distribution function(which is never zero) with an analytic function, f(x), is a quadratic polynomial. can we say that f(x) must be polynomial too? If not can you come up with a counterexample? I have to mention that we do not know if f(x) has Fourier transform or not.
thanks
This is an answer to the original version of the question, regarding the convolution of a probability distribution with an analytic function.
Say $X$ is a random variable with $P(X=0)=P(X=\pi)=0$. So the distribution of $X$ is $\mu=(\delta_0+\delta_\pi)/2$, where $\delta_t$ denotes a point mass at $t$. Let $f(t)=\sin(t)$. Then $\mu*f=0$.
In detail: $$\mu*f(x)=\int f(x-t)\,d\mu(t)=(f(x)+f(x-\pi))/2=0.$$