On a translation property of positive definite functions

Lars Omlor, and Michael Leinert. "On a translation property of positive definite functions." Banach Center Publications 89.1 (2010): 237-240. <http://eudml.org/doc/286280>.

@article{LarsOmlor2010,
abstract = {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.},
author = {Lars Omlor, Michael Leinert},
journal = {Banach Center Publications},
keywords = {IN-group; structure theorem; positive definite function; invariant neighbourhood},
language = {eng},
number = {1},
pages = {237-240},
title = {On a translation property of positive definite functions},
url = {http://eudml.org/doc/286280},
volume = {89},
year = {2010},
}

TY - JOUR
AU - Lars Omlor
AU - Michael Leinert
TI - On a translation property of positive definite functions
JO - Banach Center Publications
PY - 2010
VL - 89
IS - 1
SP - 237
EP - 240
AB - 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.
LA - eng
KW - IN-group; structure theorem; positive definite function; invariant neighbourhood
UR - http://eudml.org/doc/286280
ER -