If someone would help me understand fourier transforms i would very much appreciate it.

If we say that we have a u(x,t) where

u(0,t)=f(x)

where f(x) is some known function and the deriv w.r.t. time is zero:

(d/dt)u(0,t)=0

and we take the fourier transform in x, then how do we get a solution for u_hat in the transformed space?