## 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
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.

[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. |

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