This paper continues research done by F.H. Ruymgaart and the author. For a function f on Rd we consider its Fourier transform Ff and the functions fM(M>0) derived from Ff by the formula fM(x)=(F(εM·Ff))(−x);, where the εM are suitable integrable functions tending to 1 pointwise as M→∞. It was shown earlier that, relative to a metric dH, analogous to the Hausdorff distance between closed sets, one has dH(fM, f)=O(M−½) for all f in a certain class. We now show that, for such f, the estimate O(M−½) is optimal if and only if f has a discontinuity point.