Abstract relative Gabor transforms over canonical homogeneous spaces of semidirect product groups with Abelian normal factor

2017 ◽  
Vol 15 (06) ◽  
pp. 795-813 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Direct Product ◽  
Hilbert Function ◽  
Homogeneous Spaces ◽  
Continuous Homomorphism ◽  
Unified Approach ◽  
Locally Compact ◽  
Abstract Notion ◽  
Semidirect Product Groups ◽  
Lca Group ◽  
Hilbert Function Space

This paper introduces a unified approach to the abstract notion of relative Gabor transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let [Formula: see text] be a locally compact group, [Formula: see text] be a locally compact Abelian (LCA) group, and [Formula: see text] be a continuous homomorphism. Let [Formula: see text] be the semi-direct product of [Formula: see text] and [Formula: see text] with respect to [Formula: see text], [Formula: see text] be the canonical homogeneous space of [Formula: see text], and [Formula: see text] be the canonical relatively invariant measure on [Formula: see text]. Then we present a unified harmonic analysis approach to the theoretical aspects of the notion of relative Gabor transform over the Hilbert function space [Formula: see text].

ABSTRACT HARMONIC ANALYSIS OF RELATIVE CONVOLUTIONS OVER CANONICAL HOMOGENEOUS SPACES OF SEMIDIRECT PRODUCT GROUPS

2016 ◽  
Vol 101 (2) ◽  
pp. 171-187 ◽  
Author(s):  
ARASH GHAANI FARASHAHI
Keyword(s):  
Harmonic Analysis ◽  
Homogeneous Space ◽  
Semidirect Product ◽  
Homogeneous Spaces ◽  
Continuous Homomorphism ◽  
Compact Groups ◽  
Coset Space ◽  
Unified Approach ◽  
Left Coset ◽  
Semidirect Product Groups

This paper presents a structured study for abstract harmonic analysis of relative convolutions over canonical homogeneous spaces of semidirect product groups. Let $H,K$ be locally compact groups and $\unicode[STIX]{x1D703}:H\rightarrow \text{Aut}(K)$ be a continuous homomorphism. Let $G_{\unicode[STIX]{x1D703}}=H\ltimes _{\unicode[STIX]{x1D703}}K$ be the semidirect product of $H$ and $K$ with respect to $\unicode[STIX]{x1D703}$ and $G_{\unicode[STIX]{x1D703}}/H$ be the canonical homogeneous space (left coset space) of $G_{\unicode[STIX]{x1D703}}/H$. We present a unified approach to the harmonic analysis of relative convolutions over the canonical homogeneous space $G_{\unicode[STIX]{x1D703}}/H$.

scholarly journals Density and representation theorems for multipliers of type (p, q)

1967 ◽  
Vol 7 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Alessandro Figà-Talamanca ◽  
G. I. Gaudry
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Important Class ◽  
Lebesgue Spaces ◽  
Representation Theorems ◽  
Locally Compact ◽  
Bounded Linear ◽  
Character Group ◽  
Hausdorff Group ◽  
Lca Group

Let G be a locally compact Abelian Hausdorff group (abbreviated LCA group); let X be its character group and dx, dx be the elements of the normalised Haar measures on G and X respectively. If 1 < p, q < ∞, and Lp(G) and Lq(G) are the usual Lebesgue spaces, of index p and q respectively, with respect to dx, a multiplier of type (p, q) is defined as a bounded linear operator T from Lp(G) to Lq(G) which commutes with translations, i.e. τxT = Tτx for all x ∈ G, where τxf(y) = f(x+y). The space of multipliers of type (p, q) will be denoted by Lqp. Already, much attention has been devoted to this important class of operators (see, for example, [3], [4], [7]).

scholarly journals Ends of locally compact groups and their coset spaces

1974 ◽  
Vol 17 (3) ◽  
pp. 274-284 ◽  
Author(s):  
C. H. Houghton
Keyword(s):  
Compact Group ◽  
Direct Product ◽  
Compact Space ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Countable Base ◽  
Locally Compact ◽  
Coset Spaces ◽  
Compact Boundary

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.

scholarly journals ALGEBRA* VIOLENCE ALGEBRA CRATY

2020 ◽  
pp. 12-23
Author(s):  
Vadym Chuiko ◽  
Valerii Atamanchuk-Angel
Keyword(s):  
Sustainable Development ◽  
Direct Product ◽  
The State ◽  
State System ◽  
The Other ◽  
Philosophical Thought ◽  
Human Culture ◽  
Full Responsibility ◽  
Abstract Notion ◽  
Almost All

Almost all philosophy about the state system has concentrated on the authorities. Any function of the state can be represented as a superposition of the functions of violence / coercion. Ultimately, the state appears to be a kind of plurality of subjects with a definite crater power / coercion / violence operation on it. The algebra of trust on the multiplicity of owners of themselves, endowed with free future, is each of them is only a part of nature, еру carrier of the part of the general human culture, and for their completeness, they have and understand the need for the Other. This is the philosophy of solving political, environmental, and climate challenges not through violent / voluntaristic methods, but by the recognition of sovereign rights and the search for ways to achieve sustainable development. Any cracy / power / coercion / violence must be separated from the models of society, the state. Public agreement is not an agreement with the abstract notion of the state, but an agreement with definite elected people who have gained the trust of those to whom they temporarily render their services. Contract is temporary, limited by period, with obligatory full responsibility of the parties. Scientific novelty. For more than two thousand years, long before Aristotle and Plato, European philosophical thought, reflecting on the structure of society, wanders in the labyrinths of kratia. Modern achievements of mathematics provide an opportunity to build ideal political objects, and a direct product of material and ideal government building. (Example of a trust algebra [4].)

scholarly journals Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

2020 ◽  
Vol 20 (03) ◽  
pp. 2050015
Author(s):  
Raphaël Carroy ◽  
Andrea Medini ◽  
Sandra Müller
Keyword(s):  
Homogeneous Space ◽  
General Result ◽  
Homogeneous Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Axiom Of Determinacy ◽  
Locally Compact Spaces

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

scholarly journals Spectral mapping theorem for representations of measure algebras

1997 ◽  
Vol 40 (2) ◽  
pp. 261-266 ◽  
Author(s):  
H. Seferoǧlu
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Homomorphism ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Stieltjes Transform ◽  
Locally Compact ◽  
Unital Banach Algebra

Let G be a locally compact abelian group, M0(G) be a closed regular subalgebra of the convolution measure algebra M(G) which contains the group algebra L1(G) and ω: M0(G) → B be a continuous homomorphism of M0(G) into the unital Banach algebra B (possibly noncommutative) such that ω(L1(G)) is without order with respect to B in the sense that if for all b ∈ B, b.ω(L1(G)) = {0} implies b = 0. We prove that if sp(ω) is a synthesis set for L1(G) then the equality holds for each μ ∈ M0(G), where sp(ω) denotes the Arveson spectrum of ω, σB(.) the usual spectrum in B, the Fourier-Stieltjes transform of μ.

Nearly Countable Dense Homogeneous Spaces

2014 ◽  
Vol 66 (4) ◽  
pp. 743-758 ◽  
Author(s):  
Michael Hrušák ◽  
Jan van Mill
Keyword(s):  
Metric Spaces ◽  
Homogeneous Spaces ◽  
Structure Theorem ◽  
Polish Space ◽  
Natural Question ◽  
Borel Space ◽  
Locally Compact ◽  
Compact Spaces ◽  
Dense Sets ◽  
Locally Compact Spaces

AbstractWe study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X ∖ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or c many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.

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