scholarly journals Diffraction of Weighted Lattice Subsets

2002 ◽  
Vol 45 (4) ◽  
pp. 483-498 ◽  
Author(s):  
Michael Baake
Keyword(s):  
Abelian Group ◽  
Continuous Function ◽  
Euclidean Space ◽  
Lattice Structure ◽  
Locally Compact Abelian Group ◽  
Dual Lattice ◽  
Countable Base ◽  
Locally Compact ◽  
Bounded Complex ◽  
Dirac Comb

AbstractA Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice Γ inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniformlattice Dirac comb, and its diffraction measure is periodic, with the dual lattice Γ*as lattice of periods. This statement remains true in the setting of a locally compact Abelian group whose topology has a countable base.

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.

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).

A theorem on bounded synthesis

1971 ◽  
Vol 70 (1) ◽  
pp. 23-26
Author(s):  
E. Galanis
Keyword(s):  
Abelian Group ◽  
Continuous Function ◽  
Compact Subset ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Image Position

LetGbe a locally compact Abelian group.DEFINITION 1. A compact subset K ⊂ G is called Kroneclcer set if for every continuous function f on K of modulus identically one (|f(x)| = 1, ∀x ∈ K) and for every ε 0 there exists x ∈ Ĝ such that

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)).

Convolution Estimates and Generalized de Leeuw Theorems for Multipliers of Weak Type (1,1)

1995 ◽  
Vol 47 (2) ◽  
pp. 225-245
Author(s):  
Nakhlé Asmar ◽  
Earl Berkson ◽  
T. A. Gillespie
Keyword(s):  
Abelian Group ◽  
Weak Type ◽  
Full Range ◽  
Locally Compact Abelian Group ◽  
Strong Type ◽  
Locally Compact ◽  
Linearization Methods ◽  
Maximal Theorem ◽  
Abstract Setting ◽  
Central Result

AbstractIn the context of a locally compact abelian group, we establish maximal theorem counterparts for weak type (1,1) multipliers of the classical de Leeuw theorems for individual strong multipliers. Special methods are developed to handle the weak type (1,1) estimates involved since standard linearization methods such as Lorentz space duality do not apply to this case. In particular, our central result is a maximal theorem for convolutions with weak type (1,1) multipliers which opens avenues of approximation. These results complete a recent series of papers by the authors which extend the de Leeuw theorems to a full range of strong type and weak type maximal multiplier estimates in the abstract setting.

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