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| Pseudo-Differential Operators and $\mathfrak{T}$- Wigner Function on Locally Compact Communicative Hausdorff Groups |
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| KeyWord:Fourier transform, Wigner function, compact group, pseudo-differential operator, symbol, Weyl-Heisenberg frame |
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| Abstract: |
| In this article, we consider a harmonic analysis of locally compact groups and introduce a generalization of the classical cross-Wigner distribution defined on $G\times \hat{G}$ by
\[W_{\Im } \left(\psi ,\varphi \right)\left(g,\; \xi \right)=\int _{G}\overline{\xi \left(h\right)}\psi \left(\tau _{1} \left(g,\; h\right)\right)\overline{\varphi \left(\tau _{2} \left(g,\; h\right)\right)}d\mu \left(h\right) .\]
We construct the so-called Weyl-Heisenberg frame on a locally compact communicative Hausdorff group and establish its properties. Thus, we show that assume $\Lambda $ and $\Gamma $ are closed cocompact subgroups of $G$ and $\hat{G}$, respectively, then, for a given window $\phi \in L^{2} \left(G\right)$, either both systems $\left\{m_{\gamma } \tau _{\lambda } \phi \right\}_{\lambda \in \Lambda ,\; \gamma \in \Gamma } $ and $\left\{m_{\kappa } \tau _{\upsilon } \phi \right\}_{\kappa \in \Lambda ^{\bot } ,\; \upsilon \in \Gamma ^{\bot } } $ are Gabor systems in $L^{2} \left(G\right)$, simultaneously, with the same upper bound, or neither $\left\{m_{\gamma } \tau _{\lambda } \phi \right\}_{\lambda \in \Lambda ,\; \gamma \in \Gamma } $ nor $\left\{m_{\kappa } \tau _{\upsilon } \phi \right\}_{\kappa \in \Lambda ^{\bot } ,\; \upsilon \in \Gamma ^{\bot } } $ comprises a Gabor system. Finally, pseudo-differential operators on locally compact groups are studied, we establish that assuming a pseudo-differential operator $A_{a} $ corresponds to the symbol $a\in {\it W}_{\tau ,} {}_{1\circ \iota ^{-1} }^{\infty ,1} \left(G\times \hat{G}\right)$ then $A_{a} $ is bounded operator ${\it W}_{\tau }^{p,q} \left(G\right)\to {\it W}_{\tau }^{p,q} \left(G\right)$. |
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