I'd like to design a Digital Differentiator with a pre-specified lowpass characteristic.
I've tried Wiener Filters (using inverseFFT to transform from frequency to time-domain), Savitzky Golay and others. Intuitively, I've come to the conclusion, that by specifying DFT points, i.e. uniformly distributed samples of the frequency response, one can achieve only a maximal low-pass frequency. For example using $N$ points, the minimal lowpass frequency would be $1/N$ bins of the FFT, thus $\pi/N$ $rad/sample$.
Is this notion correct?
And further along this path: if it's true, is this a universal boundary, or are there methods, where one can implicitly or explicitly specify not-equally sampled frequency points and thus achieve lower stopband frequencies with FIR filters?
For IIR filters I think this is no problem, since with a simple $1/(z+a)$ one can choose $a$ to achieve a variable stopband frequency just as in the continuous case.