Czasopismo
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Abstrakty
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
237-240
Opis fizyczny
Daty
wydano
2010
Twórcy
autor
- Section for Computational Sensomotorics, Hertie Institute for Clinical Brain Research & Center for Integrative Neuroscience, University of Tübingen, D-72076 Tübingen, Germany
autor
- Institute of Applied Mathematics, University of Heidelberg, D-69120 Heidelberg, Germany
Bibliografia
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-15