Defining the Fourier transform of a function $f(x):\mathbb{R}\mapsto \mathbb{R}$ by: $$\hat f(k) = \int_\mathbb{R} f(x)e^{-ikx}dx.$$ In this paper on Optimized Schwarz Method, the authors mention a bounded range of frequencies $k$ when the domain is bounded and discrete. Page 19:
Since real computations are performed on bounded domains and discretized operators,the range of the frequency parameter k is not arbitrary. It is bounded from below by a lowest frequency dependent on the size of the domain in y direction and the boundary conditions imposed, $k^2> k^2_{min}$ and from above, $k$ is bounded by the mesh size $h$ in y direction, $k^2<k^2_{max}:=(\frac\pi h)^2$.
In this case, $f$ is defined in $R^2$ and we take the Fourier transform w.r.t. the variable $y$.
I can understand why $k$ is bounded from above by $\frac\pi h$, taking the discrete Fourier transform, but can't figure out this lower bound.
Can someone enlighten me on this?
Thanks.