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Fourier inversion and the Hausdorff distance
Author(s) -
Van Rooij Arnoud
Publication year - 2002
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/1467-9574.00194
Subject(s) - mathematics , pointwise , fourier transform , hausdorff distance , hausdorff space , discontinuity (linguistics) , combinatorics , class (philosophy) , locally integrable function , mathematical analysis , pure mathematics , integrable system , artificial intelligence , computer science
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.