Find all $f \in L^1(\mathbb{R})$ to satisfy $\int_{-\infty}^{\infty} f(t)e^{(-t^2+2xt)}dt = 0$

Question

The problem is

Find all $f \in L^1(\mathbb{R})$ to satisfy $\int_{-\infty}^{\infty} f(t)e^{(-t^2+2xt)}dt = 0$ a.e..

I'm studying convolution and Fourier transform in Folland chap 8. I faced this problem but I don't know at all. Someone help me?

Answer 1

The given equation is equivalent to $\int f(y)e^{-(y-x)^{2}}dy=0$ or $\int f(x-y)e^{-y^{2}}dy=0$. The convolution of $f$ with the exponential is therefore $0$ a.e.. Taking FT we see that $\hat {f} \hat {g}=0$ where $g(y)=e^{-y^{2}}$. $\hat {g}$ is never $0$ and so we get $\hat {f}=0$ which implies $f=0$ a.e. .