On a translation property of positive definite functions

• Autorzy: Lars Omlor , Michael Leinert
• Języki publikacji: EN

Abstrakty

EN

If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant $C_{h} > 0$ such that $∫ L_{x}h·g ≤ C_{h}∫ hg$ for every continuous positive definite g≥0, where $L_{x}$ is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That "for one h" implies "for all h" seemed obvious, but turned out not to be obvious at all. We fill this gap by means of a new structure theorem for IN-groups. For p ∈ ℕ even, property (*) easily implies the following property (*)ₚ: For every relatively compact invariant neighbourhood U of e, there is a constant $C_{U} > 0$ such that $||χ_{xU}·g||ₚ ≤ C_{U}||χ_{U}·g||ₚ$ for every continuous positive definite function g. For all other p ∈ (1,∞), property (*)ₚ fails (see [L]). In the special case of the unit circle, the || ||ₚ-norm results are essentially due to N. Wiener, S. Wainger, and H. S. Shapiro. For compact abelian groups they are due to M. Rains, and for locally compact abelian groups to J. Fournier.

Kategorie tematyczne

• 43-06: Proceedings, conferences, collections, etc.
• 22D05: General properties and structure of locally compact groups
• 43A35: Positive definite functions on groups, semigroups, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2010
• Tom: 89
• Numer: 1
• Strony: 237-240

Daty

wydano

2010

Twórcy

autor

Lars Omlor

• Section for Computational Sensomotorics, Hertie Institute for Clinical Brain Research & Center for Integrative Neuroscience, University of Tübingen, D-72076 Tübingen, Germany

autor

Michael Leinert

• Institute of Applied Mathematics, University of Heidelberg, D-69120 Heidelberg, Germany

Identyfikatory

DOI

10.4064/bc89-0-15