On the equivalence of the properties of quasi-translation invariance and absolute continuity of measures on a locally compact abelian group

1987 ◽  
Vol 101 (2) ◽  
pp. 279-281
Author(s):  
Denis Bell
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Simple Proof ◽  
Compact Abelian Group ◽  
Absolute Continuity ◽  
Locally Compact Abelian Group ◽  
Translation Invariance ◽  
Positive Density ◽  
Locally Compact ◽  
Translation Invariant

AbstractWe give a simple proof that a measure on a locally compact abelian group G is quasi-translation invariant with continuous translation densities if and only if it is equivalent to Haar measure on G and has a continuous positive density.

scholarly journals Multipliers for Amalgams and the Algebra S0(G)

1987 ◽  
Vol 39 (1) ◽  
pp. 123-148 ◽  
Author(s):  
Maria L. Torres De Squire
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
The Difference ◽  
Image Position

Throughout the whole paper G will be a locally compact abelian group with Haar measure m and dual group Ĝ. The difference of two sets A and B will be denoted by A ∼ B, i.e.,For a function f on G and s ∊ G, the functions f′ and fs will be defined by

On Multipliers of the Group Lp;w(G)-Algebras

2013 ◽  
Vol 59 (2) ◽  
pp. 253-268
Author(s):  
Ilker Eryilmaz ◽  
Cenap Duyar
Keyword(s):  
Abelian Group ◽  
Banach Algebra ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact

Abstract Let G be a locally compact abelian group (non-compact, non-discrete) with Haar measure and 1 ≤ p < ∞: The purpose of this paper is to study the space of multipliers on Lp;w (G) and characterize it as the algebra of all multipliers of the closely related Banach algebra of tempered elements in Lp;w (G).

scholarly journals Multipliers on some weightedLp-spaces

2000 ◽  
Vol 23 (9) ◽  
pp. 651-656
Author(s):  
S. Öztop
Keyword(s):  
Abelian Group ◽  
Weight Function ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact

LetGbe a locally compact abelian group with Haar measuredx, and letωbe a symmetric Beurling weight function onG(Reiter, 1968). In this paper, using the relations betweenpiandqi, where1<pi,   qi<∞,pi≠qi(i=1,2), we show that the space of multipliers fromLωp(G)to the spaceS(q′1,q′2,ω−1), the space of multipliers fromLωp1(G)∩Lωp2(G)toLωq(G)and the space of multipliersLωp1(G)∩Lωp2(G)toS(q′1,q′2,ω−1).

On multipliers of Fourier transforms

1972 ◽  
Vol 71 (1) ◽  
pp. 63-66 ◽  
Author(s):  
Louis Pigno
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Lebesgue Space ◽  
Fourier Transforms ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Continuous Complex ◽  
Complex Valued

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.

Multipliers for vector valued functions

1984 ◽  
Vol 99 (1-2) ◽  
pp. 137-143
Author(s):  
José Luis Torrea
Keyword(s):  
Abelian Group ◽  
Banach Spaces ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Lebesgue Spaces ◽  
Locally Compact ◽  
Translation Invariant ◽  
Bounded Linear ◽  
Vector Valued Functions ◽  
Vector Valued

SynopsisLet G be a locally compact abelian group and let Γ be the dual of G. Let A, B be Banach spaces and Lp(G,A) the Bochner-Lebesgue spaces. We prove that the space of bounded linear translation invariant operators from L1(G, A) to LX(G, B) can be identified with the space of bounded convolution invariant (in some sense) operators and also with the space of a(A, B)-valued “weak regular” measures with the relation Tf = f *μ. (A. The existence of a function m∈ L∞ (Γ,α(A,B)), such that is also proved.

Spans of translates in Lp(G)

1965 ◽  
Vol 5 (2) ◽  
pp. 216-233 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Lebesgue Space ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Linear Combinations ◽  
Character Group ◽  
Weak Closure

Throughout this paper, G denotes a Hausdorff locally compact Abelian group, X its character group, and Lp(G) (1 ≦ p ≦ ∞) the usual Lebesgue space formed relative to the Haar measure on G. If f ∈ Lp(G), we denote by Tp[f] the closure (or weak closure, if p = ∞) in Lp(G) of the set linear combinations of translates of f.

Uniqueness of uniform norm and C*-norm in L p(G, ω)

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Ibrahim Akbarbaglu ◽  
Majid Heydarpour ◽  
Saeid Maghsoudi
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Uniform Norm ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Weighted Algebra

AbstractLet G be a locally compact abelian group with a fixed Haar measure and ω be a weight on G. For 1 < p < ∞, we study uniqueness of uniform and C*-norm properties of the invariant weighted algebra L p(G, ω).

The Wiener–Pitt phenomenon on semi-groups

1963 ◽  
Vol 59 (1) ◽  
pp. 11-24 ◽  
Author(s):  
T. A. Davis
Keyword(s):  
Abelian Group ◽  
Banach Algebra ◽  
Haar Measure ◽  
Group Algebra ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Bounded Complex ◽  
Additive Measures ◽  
Complex Valued

Let G be a locally compact Abelian group, written adoptively, with Haar measure m, L1(G) the group algebra of G, and M(G) the Banach algebra of all bounded, complex-valued, regular, countably additive measures on G. For a general account of L1(G) and M(G) see Rudin (7).

Sign in / Sign up

Export Citation Format

Share Document