Integral Quantization for the Discrete Cylinder

Covariant integral quantizations are based on the resolution of identity by continuous or discrete families normalised positive operator valued measures (POVM), which have appealing probabilistic content and transform in a covariant way. One their advantages is to allow circumvent problems due presence singularities classical models. In this paper we implement for systems whose phase space $\mathbb{Z}\times\,\mathbb{S}^1$, i.e., moving circle. The symmetry group \& compact version Weyl-Heisenberg group, namely central extension abelian $\mathbb{Z}\times\,\mathrm{SO}(2)$. regard, viewed as right coset with its center. non-trivial unitary irreducible representation acting $L^2(\mathbb{S}^1)$, square integrable space. We show how derive corresponding from (weight) functions {and resulting identity}. {As particular cases latter} recover de Bi\`evre-del Olmo-Gonzales Kowalski-Rembielevski-Papaloucas coherent states Another straightforward outcome our approach Mukunda Wigner transform. also look at specific built shifted gaussians, Von Mises, Poisson, Fej\'er kernels. Applications stellar representations progress.