It is possible to create a linear operator that works as a low-pass filter, for example, the linear operator
works as a low-pass filter (I don't know if this is the proper way to write the operator), since
tends to filter the high frequencies out, while keeping the low frequencies. This can easily be realized, since if a linear filter has an impulse response that equals to
where H is the Heaviside step function, then is the operator that applies that filter, and if the filter looks like that, its Fourier transform is a sinc function, which equals to 1 for the zero frequency (the low frequencies are kept), but approaches zero as the frequency approaches +infinity.
Now, what I wonder about, is there any linear operator (kind of like the one I just showed) that applies a filter whose Fourier transform is a sigmoid function? In other words, approaches -1 when the frequency approaches -infinity, approaches +1 when the frequency approaches +infinity, and equals to 0 for the zero frequency?