It is possible to create a linear operator that works as a low-pass filter, for example, the linear operator

\hat{f} = \displaystyle{\frac{1}{2d}}\int\limits_{x-d}^{x+d}dx'

works as a low-pass filter (I don't know if this is the proper way to write the operator), since

\hat{f}\Psi(x) = \displaystyle{\frac{1}{2d}}\int\limits_{x-d}^{x+d}\Psi(x')dx'

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 \hat{f} 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?