Fourier inversion and the Hausdorff distance

This paper continues research done by F.H. Ruymgaart and the author. For a function f on R d we consider its Fourier transform Ff and the functions f M (M>0) derived from Ff by the formula f M (x) =( F(ε M · Ff))(−x);, where the ε M are suitable integrable functions tending to 1 pointwise as M→∞. It was shown earlier that, relative to a metric d H , analogous to the Hausdorff distance between closed sets, one has d H (f M , 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.