Uniform approximation by Fourier–Stieltjes transforms

1968 ◽  
Vol 64 (2) ◽  
pp. 323-333 ◽  
Author(s):  
Donald E. Ramirez
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Functions ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Norm Topology ◽  
Locally Compact ◽  
Borel Measures ◽  
Bounded Functions ◽  
Stieltjes Transforms

Let G be a locally compact Abelian group; Γ the dual group of G; CB(Γ) the algebra of continuous, bounded functions on Γ C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; M(G) the algebra of bounded Borel measures on G; M(G)^ the algebra of Fourier–Stieltjes transforms; and M(G)^− the completion of M(G)^ in the sup-norm topology on Γ. The object of this paper is to study the natural pairing between M(G)^ and M(Γ).

scholarly journals L-ideals of M(G) determined by continuity of translation

1973 ◽  
Vol 18 (4) ◽  
pp. 307-316 ◽  
Author(s):  
Gavin Brown ◽  
William Moran
Keyword(s):  
Abelian Group ◽  
Locally Compact Abelian Group ◽  
Norm Topology ◽  
Locally Compact ◽  
Total Variation Norm ◽  
Absolutely Continuous ◽  
Convolution Algebra ◽  
Borel Measures ◽  
Variation Norm ◽  
Stieltjes Transforms

G denotes a locally compact abelian group and M(G) the convolution algebra of regular bounded Borel measures on G. An ideal I of M(G) closed in the usual (total variation) norm topology is called an L-ideal if μ ∈ I, ν≪ μ (ν absolutely continuous with respect to μ) implies that ν ∈ I. Here we are concerned with the L-idealsL1(G), , and M0(G) where, as usual, L1(G) denotes the set of measures absolutely continuous with respect to Haar measure, denotes the radical of L1(G) in M(G) and M0(G) denotes the set of measures whose Fourier-Stieltjes transforms vanish at infinity.

A characterization of quotient algebras of Ll(G)

1969 ◽  
Vol 66 (3) ◽  
pp. 547-551 ◽  
Author(s):  
Donald E. Ramirez
Keyword(s):  
Continuous Functions ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Absolutely Continuous ◽  
Borel Measures ◽  
Bounded Functions ◽  
Continuous Measures ◽  
Stieltjes Transforms ◽  
Absolutely Continuous Measures

Let G be a locally compact Abelian group; Γ the dual group of G; C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; CB(Γ) the continuous, bounded functions on Γ; M (G) the algebra of bounded Borel measures on G; L1(G) the algebra of absolutely continuous measures; and M(G)∩ the algebra of Fourier–Stieltjes transforms.

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)).

scholarly journals A Theorem on Algebras of Measures on Topological Groups

1959 ◽  
Vol 11 (4) ◽  
pp. 195-206 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Topological Groups ◽  
Finite Sets ◽  
Locally Compact ◽  
Borel Measures ◽  
Bounded Complex ◽  
Image Position

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined bythe supremum being over all finite sets of disjoint Borel subsets of G.

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 Multipliers from spaces of test functions to amalgams

1993 ◽  
Vol 54 (1) ◽  
pp. 97-110
Author(s):  
Maria Torres De Squire
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Functions ◽  
Locally Compact Abelian Group ◽  
Test Functions ◽  
Space Of Continuous Functions ◽  
Locally Compact ◽  
The Fourier Transform

AbstractIn this paper we study the space of multipliers M(r, s: p, q) from the space of test functions Φrs(G), on a locally compact abelian group G, to amalgams (Lp, lq)(G); the former includes (when r = s = ∞) the space of continuous functions with compact support and the latter are extensions of the Lp(G) spaces. We prove that the space M(∞: p) is equal to the derived space (Lp)0 defined by Figá-Talamanca and give a characterization of the Fourier transform for amalgams in terms of these spaces of multipliers.

scholarly journals Constructions of the maximal strongly character invariant segal algebras and their applications

1983 ◽  
Vol 35 (1) ◽  
pp. 123-131
Author(s):  
Chang-Pao Chen
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Banach Algebras ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Segal Algebra ◽  
Segal Algebras

AbstractLet G denote any locally compact abelian group with the dual group Γ. We construct a new kind of subalgebra L1(G) ⊗ΓS of L1(G) from given Banach ideal S of L1(G). We show that L1(G) ⊗гS is the larger amoung all strongly character invariant homogeneous Banach algebras in S. when S contains a strongly character invariant Segal algebra on G, it is show that L1(G) ⊗гS is also the largest among all strongly character invariant Segal algebras in S. We give applications to characterizations of two kinds of subalgebras of L1(G)-strongly character invariant Segal algebras on G and Banach ideal in L1(G) which contain a strongly character invariant Segal algebra on G.

A diophantine problem on groups. III

1971 ◽  
Vol 70 (1) ◽  
pp. 31-47 ◽  
Author(s):  
R. C. Baker
Keyword(s):  
Abelian Group ◽  
Finite Order ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Diophantine Problem ◽  
Precise Sense ◽  
Image Position ◽  
Almost All

AbstractThe following generalization of a theorem of Weyl appeared in part I of this series of papers. Let G be a locally compact Abelian group with dual group ĝ. Let be a sequence in ĝ, not too slowly growing in a certain precise sense. Then, provided ĝ has ‘not too many’ elements of finite order, the sequencesare uniformly distributed on the circle, for almost all x in G.

scholarly journals A theorem on abstract Segal algebras over some commutative Banach algebras

1982 ◽  
Vol 25 (2) ◽  
pp. 293-301 ◽  
Author(s):  
U.B. Tewari ◽  
K. Parthasarathy
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Compact Abelian Group ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Ideal Space ◽  
Locally Compact ◽  
Commutative Banach Algebras

Let B be a commutative, semi-simple, regular, Tauberian Banach algebra with noncompact maximal ideal space Δ(B). Suppose B has the property that there is a constant C such that for every compact subset K of Δ(B) there exists a f ∈ B with = 1 on K, ‖f‖B ≤ C and has compact support. We prove that if A is a proper abstract Segal algebra over B then for every positive integer n there exists f ∈ B such that fn ∉ A but fn+1 ∈ A. As a consequence of this result we prove that if G is a nondiscrete locally compact abelian group, μ a positive unbounded Radon measure on Γ (the dual group of G), 1 ≤ p < q < ∞ and , then .

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