This paper presents an in-depth analysis of the square integrable representations of the generalized Weyl-Heisenberg group. We conduct a thorough investigation into the quasi-regular representation of this group, examining its properties and implications. Additionally, we characterize the admissible conditions using the Fourier transform. To support our theoretical findings, we provide several examples that illustrate and validate our technical considerations.

Fatemeh Esmaeelzadeh is an associate professor at the Islamic Azad University, Bojnourd Branch (IAU), in Iran. She received her PhD, M.Sc, and B.Sc degrees in pure mathematics from Ferdowsi University of Mashhad in 2012, 2002, and 1999, respectively. Her research interests include abstract harmonic analysis, wavelet and frame theory, representation theory on topological groups and homogeneous spaces, and localization operators. Dr. Esmaeelzadeh has published approximately 40 scientific articles in peer-reviewed ISI journals and conference proceedings. She currently serves as an associate professor in the mathematics department at IAU, Bojnourd Branch. She is an advisor to two PhD students and supervises one MSc student. Additionally, Dr. Esmaeelzadeh has been invited as a reviewer by several peer-reviewed journals, including Wavelet and Linear Algebra, Sahand Communications in Mathematical Analysis, Journal of Linear and Topological Algebra, and others. She is also a member of the committee on the defense of PhD and M.Sc dissertations. In 2014, Dr. Esmaeelzadeh visited the Numerical Harmonic Analysis Group (NuHAG) at the University of Vienna in Austria. In 2012, she attended the Joint ICTP-TWAS School in Italy.