On a certain invariant of a locally compact group

Leptin, Horst

doi:10.1090/s0002-9904-1966-11594-x

Cited by 9 publications

(5 citation statements)

References 6 publications

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“…We will define a certain uniform density that is based on the group invariant I (G) introduced by Leptin [38], see also [50,Cor. 4.14].…”

Section: Leptin Density Of a Measurementioning

confidence: 99%

“…This article addresses asymptotic frequencies of point sets in locally compact groups G that are amenable [50]. 1 We introduce a canonical notion of uniform density, which is intimately related to the group invariant I (G) introduced by Horst Leptin in [38], and we thus call it the Leptin density of a locally finite point set or more generally of a positive Borel measure on G. Let us restrict to amenable locally compact groups that are unimodular. This setting encompasses locally compact groups that are exponentially bounded [50,Proposition 6.8], such as locally compact abelian groups.…”

Section: Introductionmentioning

confidence: 99%

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Leptin Densities in Amenable Groups

Pogorzelski

Richard

Strungaru

2022

J Fourier Anal Appl

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Consider a positive Borel measure on a locally compact group. We define a notion of uniform density for such a measure, which is based on a group invariant introduced by Leptin in 1966. We then restrict to unimodular amenable groups and to translation bounded measures. In that case our density notion coincides with the well-known Beurling density from Fourier analysis, also known as Banach density from dynamical systems theory. We use Leptin densities for a geometric proof of the model set density formula, which expresses the density of a uniform regular model set in terms of the volume of its window, and for a proof of uniform mean almost periodicity of such model sets.

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“…We will define a certain uniform density that is based on the group invariant I (G) introduced by Leptin [38], see also [50,Cor. 4.14].…”

Section: Leptin Density Of a Measurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Leptin Densities in Amenable Groups

Pogorzelski

Richard

Strungaru

2022

J Fourier Anal Appl

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“…We will define a certain uniform density that is based on the group invariant IpGq introduced by Leptin [31], see also [42,Cor. 4.14].…”

Section: Leptin Densities Of Positive Measuresmentioning

confidence: 99%

“…This article addresses asymptotic frequencies of infinite point sets in unimodular amenable locally compact Hausdorff groups. We introduce a canonical notion of uniform density, which is intimately related to Leptin's group invariant IpGq defined in [31], and we thus call it the Leptin density of a point set or more generally of a translation bounded measure on G. We show that Leptin densities coincide with Banach densities. Our definition of Leptin density for amenable groups is inspired by work of Gröchenig, Kutyniok and Seip [19], who observed its usefulness for sampling and interpolation problems in locally compact abelian groups, compare [44,Lem.…”

Section: Introductionmentioning

confidence: 99%

Leptin densities in amenable groups

Pogorzelski¹,

Richard²,

Strungaru³

2021

Preprint

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We define a notion of uniform density on translation bounded measures in unimodular amenable locally compact Hausdorff groups, which is based on a group invariant introduced by Leptin in 1966. We show that this density notion coincides with the well-known Banach density. We use Leptin densities for a new geometric proof of the model set density formula, which expresses the density of a uniform regular model set in terms of the volume of its window, and for a proof of uniform mean almost periodicity of such model sets.

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“…Using his proof [4] of the L -conjecture (p > 2) for discrete groups, and a powerful reduction theorem, Rajagopalan [5] proved the L -conjecture (p > 2) . In 1966 Leptin [3] introduced a two-valued function I{G) , and proved the L -conjecture among groups G with It is well known (see for example Hewitt and Ross [7]) that compactness of G implies that Lp(G) is closed under convolution, for all p -1 . The following proof of the converse, for p > 2 , proceeds from first principles.…”

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confidence: 99%

An elementary proof of part of a classical conjecture

Gaudet

Gamlen

1970

Bull. Austral. Math. Soc.

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In this paper we present a new proof of the L -conjecture for p > 2 which is elementary and much shorter than that of Rajagopalan [5]. If G is a locally compact Hausdorff group with left Haar measure m , we define convolution as follows: If / and g are measurable functions for which the integral f(y)ff(.y x)dm(y) exists ae[m] , we >G say f*g exists and we define f*g{x) = f(y)g[y~ x)dm(y) if this >G integral exists and f*g(x) = 0 otherwise. As usual, L {G)

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On a certain invariant of a locally compact group

Cited by 9 publications

References 6 publications

Leptin Densities in Amenable Groups

Leptin Densities in Amenable Groups

Leptin densities in amenable groups

An elementary proof of part of a classical conjecture

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