15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 7 - Funktionalanalysis, Harmonische Analysis
Donnerstag, 20. September 2001, 14.30, Hörsaal 28

 

Extension of a theorem of Wiener to IN-groups

Michael Leinert, Universität Heidelberg

 

Wiener has shown that an integrable function on the circle $\textbf{T}$ which is square integrable near the identity and has nonnegative Fourier transform, is square integrable on all of $\textbf{T}$. 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 $p$-integrable in place of square integrable holds for all even $p$ and fails for all other $p \in [1,\infty)$ 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 $p \in [1,\infty)$ in the case of the $ax + b$-group.

[1]	J. J. F. Fournier, Local and global properties of functions and their Fourier transforms, Tôhoku Math. J. 49 (1997), 115-131.
[2]	M. Leinert, On a theorem of Wiener, submitted.
[3]	H. S. Shapiro, Majorant problems for Fourier coefficients, Quart. J. Math. Oxford (2) 26 (1975), 9-18.

E-Mail:

leinert@math.uni-heidelberg.de

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