Michael Leinert, Universität Heidelberg
Wiener has shown that an integrable function on the circle which is square integrable near the identity and has nonnegative Fourier transform, is square integrable on all of . In the last 30 years this has been extended by the work of various authors step by step. The latest result, which is due to Fournier, states that, in a suitable reformulation, Wiener's theorem with -integrable in place of square integrable holds for all even and fails for all other in the case of a general locally compact abelian group. We extend this to all IN-groups (locally compact groups with at least one invariant compact neighbourhood of the identity) and show that an extension to all locally compact groups is not possible: Wiener's theorem fails for all in the case of the -group.
|||J. J. F. Fournier, Local and global properties of functions and their Fourier transforms, Tôhoku Math. J. 49 (1997), 115-131.|
|||M. Leinert, On a theorem of Wiener, submitted.|
|||H. S. Shapiro, Majorant problems for Fourier coefficients, Quart. J. Math. Oxford (2) 26 (1975), 9-18.|