Lax-Like Stability for the Discretization of Pseudodifferential Operators through Gabor Multipliers and Spline-Type Spaces

In this paper we study the stability of projection schemes for pseudodifferential operators defined over a locally compact Abelian (LCA) group G unto a space of generalized Gabor multipliers (GGM), also called time-frequency multipliers. The projection is reformulated as a projection of the symbol operator into the spline-type (ST) space generated by the Rihaczek distributions that characterize the selected space of multipliers and the related subgroup of the time-frequency space G×G. The symplectic nature of the time-frequency group is avoided, hence a constructive realizable algorithm can be performed on the LCA group G × G. Stability is defined as uniform boundedness of a sequence of projections induced by an automorphism over the group G. We will describe the automorphisms that generate a sequence of GGM spaces and the ones that characterize stability.