I am looking for square integratable if possible smooth test functions $f(x)$ such that it is zero for $x<A$. On top of this I want to have the property that the real part of their Fourier transform $\hat{f}(k)$ for large values of $k$ decay at least like $\sim e^{-ak^2}$.
Do such functions exist and if so how can I construct one? My first guess was trying to convolute the function $f(x)=x^2e^{-x^2}$ with an heavy side function. This function f(x) has the right decaying behaviour but combined with the heavy side function it does not.