Comment by kalal

My background is in image processing and I can see quite clearly that the high frequency noise happens when you subtract two images, one of which is shifted by half-pixel for instance. What you are left with is the edges (high frequency) of the image. The same applies to 1D in my opinion. For this reason I am not convinced (1) is actually true. The argument about lack of low frequency is interesting, who knows what is happening there indeed.