The evolution of a pulse can be described by

y(x,t) = Re((1/(sqrt (2 pi)) * integral ( phi (k) exp [i(kx-wt)] dk )


phi (k) = 1/(sqrt (2 pi)) * integral ( y(x,t=0) exp [-ikx] dx )

Suppose phi(k) is given by

phi (k) = phi_o when abs(k-k_o) is less than or equal to delta k and 0 everywhere else

Determine y(x,t=0) symbolically

I don't know where to start. I first thought that I should start with the y(x,t) equation, plug in phi_o and try to integrate from abs (k-k_o) to delta k, but the dk with the y(x,t) is confusing to me.
Could you please help to get me started?