Numerical stability of fft/ifft algorithm

Short: The IFFT of a function matches an analytic solution if sampling points in spectral space (k) "reasonable", but will yield an exponential growth if k becomes large. Is there numerical stability restriction on the IFFT algorithm?

Long: The code I am writing takes a force in physical space, F(x,z) which is defined over a domain L and with a sampling separation dx, transforms it to spectral space (x transforms to wavenumber k, z is not transformed), uses the result to construct another function, u(k,z), and then transforms the new function back to physical space, U(x,z). Here is a snippet of pseudocode:

x = array of uniformly spaced points from -2L to 2L ; 
z = array of logarithmically spaced points from z0 to 3L; 
Nx = number of points in x direction; 
Nz = number of points in z direction;

dk = 2*pi/dx;
k = dk*(0:Nx/2)/Nx;
Nk = Nx/2+1; 

f = f(x,z); %function in physical space 
f_tilda = fft(f);
f_tilda = f_tilda(1:Nx/2+1,:)*dx/(2*pi);

u_tilda = u_tilda(k,z); %function in spectral space, evaluated over positive k 
u_tilda = cat(1, u_tilda, conj(u_tilda(end-1:-1:2,:))); %u_tilda is known to be hermitian

u = real(ifft(u_tilda)*(2*pi/dx));

I have an analytic solution to compare the code against, and what I am finding is that if I choose L and dx "properly" the code matches the analytic solution reasonably well. But if dx taken too small or L taken too small, the result is exponential growth.

Could this be a problem with numerical stability of the IFFT/FFT algorithm?