scholarly journals Remarks on the paper “Transient Markov convolution semi-groups and the associated negative definite functions”

1986 ◽  
Vol 102 ◽  
pp. 181-184 ◽  
Author(s):  
Masayuki Itô
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Definite Function ◽  
Semi Group ◽  
Locally Compact ◽  
Negative Definite Function

Let X be a locally compact and σ-compact abelian group and let denote the dual group of X. We denote by ξ a fixed Haar measure on X and by the Haar measure associated with ξ. In [2], we show the followingTheorem. Let (αt)t≧0 be a sub-Markov convolution semi-group on X and let ψ be the negative definite function associated with (αt)t≧0. Then (αt)t≧0 is transient if and only if Re (1/ψ) is locally -summable.

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

scholarly journals Transient Markov convolution semi-groups and the associated negative definite functions

1983 ◽  
Vol 92 ◽  
pp. 153-161 ◽  
Author(s):  
Masayuki Itô
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact

Let X be a locally compact and σ-compact abelian group and be the dual group of X. We denote by ξ a fixed Haar measure on X and by the Haar measure on associated with ξ.

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

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).

Gap Series on Groups and Spheres

1966 ◽  
Vol 18 ◽  
pp. 389-398 ◽  
Author(s):  
Daniel Rider
Keyword(s):  
Abelian Group ◽  
Trigonometric Polynomial ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Gap Series ◽  
Image Position

Let G be a compact abelian group and E a subset of its dual group Γ. A function ƒ ∈ L1(G) is called an E-function if for all γ ∉ E wheredx is the Haar measure on G. A trigonometric polynomial that is also an E-function is called an E-polynomial.

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.

On the Upper Majorant Property for Locally Compact Abelian Groups

1978 ◽  
Vol 30 (5) ◽  
pp. 915-925
Author(s):  
M. Rains
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Let G be a compact abelian group and form the spaces LP(G) with respect to the normalized Haar measure on G.

scholarly journals Characterization of Relative Domination Principle

1973 ◽  
Vol 50 ◽  
pp. 175-184 ◽  
Author(s):  
Isao Higuchi ◽  
Masayuki Itô
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Radon Measure ◽  
Convolution Type ◽  
Convolution Kernel ◽  
Locally Compact ◽  
Positive Radon Measure ◽  
Convolution Kernels

Let X be a locally compact and σ-compact Abelian group and ξ be the Haar measure of X. A positive Radon measure N on X is called a convolution kernel when we regard it as a kernel of potentials of convolution type. M. Itô [4], [6] characterized the convolution kernel which satisfies the domination principle. The purpose of this paper is to characterize the relative domination principle for the convolution kernels.

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