The aim of this thesis is to investigate generalisations of common quantization schemes both in the continuous and in the discrete setting. Precisely, we study generalisations of Weyl quantised pseudo-differential operators, defined on Rn, to pseudo-differential oper- ators where the spatial variable of the symbol depends on a non-linear function. Results of this kind are, at least to the author, unknown in the literature, despite the fact that some generalizations of the Weyl quantization have been recently studied by Toft. The interest in the generalization studied herein relies on its links to a feasible definition of a Weyl quantization on the Heisenberg group.
In the second part of the work we study Weyl quantization for pseudo-difference opera- tors, i.e. operators with symbols having spatial variable on the lattice Zn. The work can be considered as a natural continuation of the work of Ruzhansky, Botchway and Kibiti. Its importance lies in the interconnections it has with the Weyl quantization on the torus, which is a meaningful issue to be asked.