Suppose I have some function $f(t,x)$ which obeys some second-order PDE. Suppose further that the initial conditions $$ f(0,x) = 0 \qquad \text{and} \qquad \partial_t f(0,x) =0 $$ are satisfied.
What do the initial conditions mean in Fourier space? That is to say, if I consider the transform $F(\omega,x)$ where $$ f(t, x) \ = \ \int_{-\infty}^{\infty} d\omega\ e^{- 2 \pi i \omega t} F(\omega,x) \ . $$ It seems that the initial conditions imply $$ 0 \ = \ \int_{-\infty}^{\infty} d\omega\ F(\omega,x) $$ as well as $$ 0 \ = \ \int_{-\infty}^{\infty} d\omega\ ( - 2 \pi i ) \omega F(\omega,x) \ . $$ This seems to me like all this tells me is that the function $F(\omega,x)$ must be odd - can anything else be inferred from the initial conditions in Fourier space?