Question: weird plot output

Hi, I was trying to define a function defined as a Fourier transform. In the plot, there are several pieces of the curve missing.

> f0 := (k) -> sin((1/2)*k)/(2*Pi-k) - sin((1/2)*k)/(2*Pi+k) + 2*sin((1/2)*k)/k ;
> gauss1 := (q, k)  -> exp( -(1/2)*q^2/.2^2 - (1/200)*k^2 ) ;
> gauss2 := (q, k)  -> exp( -(1/200)*q^2 - (1/2)*k^2/.2^2 ) ;
> F1 := (x)  -> evalf(Int( f0(-k) * gauss1(0, k) * gauss2((1/2)*k, 0) * cos(x*k) , k = -infinity .. infinity)) ;
> F2 := (x)  -> evalf(Int( f0(-k) * gauss1(0, k) * cos(x*k),  k = -infinity .. infinity)) ;
> F3 := (x)  -> evalf(Int( ( sin(Pi*(x-y)) )^2 * exp( -y^2* (1/(2*.1^2)+16/(2*100)) ), y = x-1 .. x));
> plot(F1(t), t = -2 .. 2);
 

I know the function F1 equals F3 (apart from a shift and a factor), which is a convolution. F2 is slightly larger than F1, but has
the same shape
The plot of F3 is just fine.
Does anybody have an idea of what could be going on?
Maple calculates the values missing from the plot, e.g. F1(0),F1(0.5) just fine, but very slow.
The plot of F1, F2 requires several minutes, the plot of F3 less than one minute. 
I am left guessing. Any help would be appreciated.

Plot of F1

Plot of F2(x)


Plot of F3(x+1/2)