caa a22 4500 606200738 CHVBK 20210128100945.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s00025-014-0407-1 doi (NATIONALLICENCE)springer-10.1007/s00025-014-0407-1 Ghaani Farashahi Arash Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna, Vienna, Austria aut A Unified Group Theoretical Method for the Partial Fourier Analysis on Semi-Direct Product of Locally Compact Groups [Elektronische Daten] [Arash Ghaani Farashahi] Let H and K be locally compact groups and $${\tau : H \to Aut(K)}$$ τ : H → A u t ( K ) be a continuous homomorphism. Further let $${G_\tau = H \ltimes_\tau K}$$ G τ = H ⋉ τ K be the semi-direct product of H and K with respect to the continuous homomorphism $${\tau}$$ τ . This paper presents a unified approach for the partial Fourier analysis on $${G_\tau = H \ltimes_\tau K}$$ G τ = H ⋉ τ K , when K is Abelian. The $${\tau}$$ τ -dual group (partial dual group) $${G_{\widehat{\tau}}}$$ G τ ^ of $${G_\tau}$$ G τ is defined as the semi-direct product group $${H \ltimes_{\widehat{\tau}}\widehat{K}}$$ H ⋉ τ ^ K ^ , where $${\widehat{\tau}: H \to Aut(\widehat{K})}$$ τ ^ : H → A u t ( K ^ ) is given via $${\widehat{\tau}_h(\omega) : = \omega \circ \tau_{h^{-1}}}$$ τ ^ h ( ω ) : = ω ∘ τ h - 1 for all $${h \in H}$$ h ∈ H and $${\omega \in \widehat{K}}$$ ω ∈ K ^ . We will prove a Pontrjagin duality theorem and we introduce a unitary partial Fourier transform on $${G_\tau}$$ G τ . As examples, we shall study these techniques for some well-known semi-direct product groups. Springer Basel, 2014 Semi-direct products of locally compact groups nationallicence partial Fourier transform nationallicence character group nationallicence $${\tau}$$ τ -dual group nationallicence Pontrjagin duality nationallicence $${\tau}$$ τ -Fourier transform nationallicence Parseval formula nationallicence Plancherel Theorem nationallicence inversion formula nationallicence Results in Mathematics Springer Basel 67/1-2(2015-02-01), 235-251 1422-6383 67:1-2<235 2015 67 25 https://doi.org/10.1007/s00025-014-0407-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00025-014-0407-1 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Ghaani Farashahi Arash Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna, Vienna, Austria aut NATIONALLICENCE 773 0- Results in Mathematics Springer Basel 67/1-2(2015-02-01), 235-251 1422-6383 67:1-2<235 2015 67 25