If:

Then would the inverse Fourier Transform be:

I think so because Mathematica can calculate inverse transforms. I used the code below with the indicated values for the parameters:

Code:

In[23]:= \[Rho] = 1;
w = 3;
f1 = 1;
p[f_] := Piecewise[
{{(1/2)*w, Abs[f] < f1},
{(1/(4*w))*(1 + Cos[Pi/(2*w*\[Rho])]*
(Abs[f] - w*(1 - \[Rho]))),
f1 <= Abs[f] <= 2*w - f1},
{0, Abs[f] > 2*w - f1}}];
FullSimplify[InverseFourierTransform[
p[f], f, x]]

and obtained:

and this is the same results obtained by the integration above.