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Arash Ghaani Farashahi - One of the best experts on this subject based on the ideXlab platform.

  • a unified group theoretical method for the partial fourier analysis on semi direct product of locally compact groups
    Results in Mathematics, 2015
    Co-Authors: Arash Ghaani Farashahi
    Abstract:

    Let H and K be locally compact groups and \({\tau : H \to Aut(K)}\) be a Continuous Homomorphism. Further let \({G_\tau = H \ltimes_\tau K}\) be the semi-direct product of H and K with respect to the Continuous Homomorphism \({\tau}\). This paper presents a unified approach for the partial Fourier analysis on \({G_\tau = H \ltimes_\tau K}\), when K is Abelian. The \({\tau}\)-dual group (partial dual group) \({G_{\widehat{\tau}}}\) of \({G_\tau}\) is defined as the semi-direct product group \({H \ltimes_{\widehat{\tau}}\widehat{K}}\), where \({\widehat{\tau}: H \to Aut(\widehat{K})}\) is given via \({\widehat{\tau}_h(\omega) : = \omega \circ \tau_{h^{-1}}}\) for all \({h \in H}\) and \({\omega \in \widehat{K}}\). We will prove a Pontrjagin duality theorem and we introduce a unitary partial Fourier transform on \({G_\tau}\). As examples, we shall study these techniques for some well-known semi-direct product groups.

  • Continuous Partial Gabor Transform for Semi-Direct Product of Locally Compact Groups
    Bulletin of the Malaysian Mathematical Sciences Society, 2014
    Co-Authors: Arash Ghaani Farashahi
    Abstract:

    Let \(H\) be a locally compact group, \(K\) be an LCA group, \(\tau :H\rightarrow Aut(K)\) be a Continuous Homomorphism and \(G_\tau =H\ltimes _\tau K\) be the semi-direct product of \(H\) and \(K\) with respect to the Continuous Homomorphism \(\tau \). In this article, we introduce the \(\tau \times \widehat{\tau }\)-time frequency group \(G_{\tau \times \widehat{\tau }}\). We define the \(\tau \times \widehat{\tau }\)-Continuous Gabor transform of \(f\in L^2(G_\tau )\) with respect to a window function \(u\in L^2(K)\) as a function defined on \(G_{\tau \times \widehat{\tau }}\). It is also shown that the \(\tau \times \widehat{\tau }\)-Continuous Gabor transform satisfies the Plancherel Theorem and reconstruction formula. This approach is tailored for choosing elements of \(L^2(G_\tau )\) as a window function. Finally, we indicate some possible applications of these methods in the case of some well-known semi-direct product groups.

  • Zak Transform for Semidirect Product of Locally Compact Groups
    arXiv: Functional Analysis, 2012
    Co-Authors: Arash Ghaani Farashahi, Ali Akbar Arefijamaal
    Abstract:

    Let $H$ be a locally compact group and $K$ be an LCA group also let $\tau:H\to Aut(K)$ be a Continuous Homomorphism and $G_\tau=H\ltimes_\tau K$ be the semidirect product of $H$ and $K$ with respect to $\tau$. In this article we define the Zak transform $\mathcal{Z}_L$ on $L^2(G_\tau)$ with respect to a $\tau$-invariant uniform lattice $L$ of $K$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we show that how these techniques apply for the semidirect product group $\mathrm{SL}(2,\mathbb{Z})\ltimes_\tau\mathbb{R}^2$ and also the Weyl-Heisenberg groups.

  • Continuous Gabor transform for semi-direct product of locally compact groups
    arXiv: Functional Analysis, 2012
    Co-Authors: Arash Ghaani Farashahi
    Abstract:

    Let $H$ be a locally compact group, $K$ be an LCA group, $\tau:H\to Aut(K)$ be a Continuous Homomorphism and $G_\tau=H\ltimes_\tau K$ be the semi-direct product of $H$ and $K$ with respect to the Continuous Homomorphism $\tau$. In this article we introduce the $\tau\times\hat{\tau}$-time frequency group $G_{\tau\times\hat{\tau}}$. We define the $\tau\times\hat{\tau}$-Continuous Gabor transform of $f\in L^2(G_\tau)$ with respect to a window function $u\in L^2(K)$ as a function defined on $G_{\tau\times\hat{\tau}}$. It is also shown that the $\tau\times\hat{\tau}$-Continuous Gabor transform satisfies the Plancherel Theorem and reconstruction formula. This approach is tailored for choosing elements of $L^2(G_\tau)$ as a window function. Finally, we illustrate application of these methods in the case of some well-known semi-direct product groups.

  • A new approach to the Fourier analysis on semi-direct products of groups
    arXiv: Functional Analysis, 2012
    Co-Authors: Arash Ghaani Farashahi
    Abstract:

    Let $H$ and $K$ be locally compact groups and also $\tau:H\to Aut(K)$ be a Continuous Homomorphism and $G_\tau=H\ltimes_\tau K$ be the semi-direct product of $H$ and $K$ with respect to the Continuous Homomorphism $\tau$. This paper presents a novel approach to the Fourier analysis of $G_\tau$, when $K$ is abelian. We define the $\tau$-dual group $G_{\hat{\tau}}$ of $G_\tau$ as the semi-direct product $H\ltimes_{\hat{\tau}}\hat{K}$, where $\hat{\tau}:H\to Aut(\hat{K})$ defined via (\ref{A}). We prove a Ponterjagin duality Theorem and also we study $\tau$-Fourier transforms on $G_\tau$. As a concrete application we show that how these techniques apply for the affine group and also we compute the $\tau$-dual group of Euclidean groups and the Weyl-Heisenberg groups.

Mikhail Tkachenko - One of the best experts on this subject based on the ideXlab platform.

  • Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids
    Axioms, 2020
    Co-Authors: Mikhail Tkachenko
    Abstract:

    We study the factorization properties of Continuous Homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = ∏ i ∈ I D i of topological or even topologized monoids. In a number of different situations, we establish that every Continuous Homomorphism f : S → K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every Continuous Homomorphism f of S to K has a finite type, which means that f admits a Continuous factorization through a finite subproduct of D. A similar conclusion is obtained for Continuous Homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results.

  • Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids
    Axioms, 2019
    Co-Authors: Mikhail Tkachenko
    Abstract:

    We study factorization properties of Continuous Homomorphisms defined on submonoids of products of topologized monoids. We prove that if S is an ω-retractable submonoid of a product D = ∏ i ∈ I D i of topologized monoids and f : S → H is a Continuous Homomorphism to a topologized semigroup H with ψ ( H ) ≤ ω , then one can find a countable subset E of I and a Continuous Homomorphism g : p E ( S ) → H satisfying f = g ∘ p E ↾ S , where p E is the projection of D to ∏ i ∈ E D i . The same conclusion is valid if S contains the Σ -product Σ D ⊂ D . Furthermore, we show that in both cases, there exists the smallest by inclusion subset E ⊂ I with the aforementioned properties.

  • Feebly compact paratopological groups and real-valued functions
    Monatshefte für Mathematik, 2012
    Co-Authors: Manuel Sanchis, Mikhail Tkachenko
    Abstract:

    We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with Continuous multiplication) which provide answers to a number of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group G can fail to be a topological group. Our group G has the Baire property, is Fréchet–Urysohn, but it is not precompact. It is well known that every infinite pseudocompact topological group contains a countable non-closed subset. We construct an infinite feebly compact Hausdorff paratopological group G all countable subsets of which are closed. Another peculiarity of the group G is that it contains a nonempty open subsemigroup C such that C−1 is closed and discrete, i.e., the inversion in G is extremely disContinuous. We also prove that for every Continuous real-valued function g on a feebly compact paratopological group G , one can find a Continuous Homomorphism φ of G onto a second countable Hausdorff topological group H and a Continuous real-valued function h on H such that g=h∘φ . In particular, every feebly compact paratopological group is R3 -factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups

  • Feebly compact paratopological groups and real-valued functions
    Monatshefte für Mathematik, 2012
    Co-Authors: Manuel Sanchis, Mikhail Tkachenko
    Abstract:

    We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with Continuous multiplication) which provide answers to a num- ber of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group G can fail to be a topological group. Our group G has the Baire property, is Fr´ echet-Urysohn, but it is not precompact. It is well known that every infinite pseudocompact topological group contains a countable non-closed subset. We construct an infinite feebly compact Hausdorff paratopological group G all countable subsets of which are closed. Another peculiar- ity of the group G is that it contains a nonempty open subsemigroup C such that C 1 is closed and discrete, i.e., the inversion in G is extremely disContinuous. We also prove that for every Continuous real-valued function g on a feebly com- pact paratopological group G, one can find a Continuous Homomorphism j of G onto a second countable Hausdorff topological group H and a Continuous real-valued func- tion h on H such that g = h j. In particular, every feebly compact paratopological group is R3-factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups.

Linus Kramer - One of the best experts on this subject based on the ideXlab platform.

Richard Bödi - One of the best experts on this subject based on the ideXlab platform.

Taras Banakh - One of the best experts on this subject based on the ideXlab platform.

  • On images of complete topologized subsemilattices in sequential semitopological semilattices
    Semigroup Forum, 2019
    Co-Authors: Taras Banakh, Serhii Bardyla
    Abstract:

    A topologized semilattice X is called complete if each non-empty chain $$C\subset X$$ C ⊂ X has $$\inf C\in {\bar{C}}$$ inf C ∈ C ¯ and $$\sup C\in {\bar{C}}$$ sup C ∈ C ¯ . We prove that for any Continuous Homomorphism $$h:X\rightarrow Y$$ h : X → Y from a complete topologized semilattice X to a sequential Hausdorff semitopological semilattice Y the image h ( X ) is closed in Y .

  • Characterizing chain-compact and chain-finite topological semilattices
    Semigroup Forum, 2019
    Co-Authors: Taras Banakh, Serhii Bardyla
    Abstract:

    In the paper we present various characterizations of chain-compact and chain-finite topological semilattices. A topological semilattice X is called chain-compact (resp. chain-finite ) if each closed chain in X is compact (finite). In particular, we prove that a (Hausdorff) $$T_1$$ T 1 -topological semilattice X is chain-finite (chain-compact) if and only if for any closed subsemilattice $$Z\subset X$$ Z ⊂ X and any Continuous Homomorphism $$h:Z\rightarrow Y$$ h : Z → Y to a (Hausdorff) $$T_1$$ T 1 -topological semilattice Y the image h ( Z ) is closed in Y .

  • Completeness and absolute H-closedness of topological semilattices
    Topology and its Applications, 2019
    Co-Authors: Taras Banakh, Serhii Bardyla
    Abstract:

    Abstract We find (completeness type) conditions on topological semilattices X , Y guaranteeing that each Continuous Homomorphism h : X → Y has closed image h ( X ) in Y.

  • A simple inductive proof of Levy-Steinitz theorem
    arXiv: Functional Analysis, 2017
    Co-Authors: Taras Banakh
    Abstract:

    We present a relatively simple inductive proof of the classical Levy-Steinitz Theorem saying that for a sequence $(x_n)_{n=1}^\infty$ in a finite-dimensional Banach space $X$ the set of all sums of rearranged series $\sum_{n=1}^\infty x_{\sigma(n)}$ is an affine subspace of $X$. This affine subspace is not empty if and only if for any linear functional $f:X\to \mathbb R$ the series $\sum_{n=1}^\infty f(x_{\sigma(n)})$ is convergent for some permutation $\sigma$ of $\mathbb N$. This gives an answer to a problem of Vaja Tarieladze, posed in Lviv Scottish Book in September, 2017. Also we construct a sequence $(x_n)_{n=1}^\infty$ in the torus $\mathbb T\times\mathbb T$ such that the series $\sum_{n=1}^\infty x_{\sigma(n)}$ is divergent for all permutations $\sigma$ of $\mathbb N$ but for any Continuous Homomorphism $f:\mathbb T^2\to\mathbb T$ to the circle group $\mathbb T:=\mathbb R/\mathbb Z$ the series $\sum_{n=1}^\infty f(x_{\sigma_f(n)})$ is convergent for some permutation $\sigma_f$ of $\mathbb N$. This example shows that the second part of Levy-Steinitz Theorem (characterizing sequences with non-empty set of potential sums) does not extend to locally compact Abelian groups.