TY - JOUR
T1 - Pseudo-Differential Operators and $\mathfrak{T}$- Wigner Function on Locally Compact Communicative Hausdorff Groups
AU - Yaremenko , M.I.
JO - Journal of Nonlinear Modeling and Analysis
VL - 4
SP - 1031
EP - 1045
PY - 2024
DA - 2024/12
SN - 6
DO - http://doi.org/10.12150/jnma.2024.1031
UR - https://global-sci.org/intro/article_detail/jnma/23670.html
KW - Fourier transform, Wigner function, compact group, pseudo-differential operator, symbol, Weyl-Heisenberg frame.
AB - <p style="text-align: justify;">In this article, we consider a harmonic analysis of locally compact
groups and introduce a generalization of the classical cross-Wigner distribution
defined on $G × \hat{G}$ by $$W_{\mathfrak{F}}(\psi,\varphi)(g,\xi)=\int_{G}\overline{\xi(h)}\psi(\tau_1(g,h))\overline{\varphi(\tau_2(g,h))}d\mu(h).$$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 $\phi∈ L^2(G),$ either both systems $\{m_{\gamma}\tau_{\lambda}\phi\}_{\lambda\in\Lambda,\gamma\in\Gamma}$&nbsp;and $\{m_{\kappa}\tau_v\phi\}_{\kappa\in\Lambda^{⊥},v\in\Gamma^{⊥}}$&nbsp;are Gabor systems in $L^2
(G),$ simultaneously, with
the same upper bound, or neither&nbsp;$\{m_{\gamma}\tau_{\lambda}\phi\}_{\lambda\in\Lambda,\gamma\in\Gamma}$ nor&nbsp;$\{m_{\kappa}\tau_v\phi\}_{\kappa\in\Lambda^{⊥},v\in\Gamma^{⊥}}$&nbsp;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 $a\in W^{\infty,1}_{\tau,1oι^{-1}}(G\times\hat{G})$&nbsp;then $A_a$ is
bounded operator $W^{p,q}_{\tau}(G)\rightarrow W^{p,q}_{\tau}(G)$.</p>