Semi-Discrete Sampling in Sobolev-Orlicz Spaces

In this paper, we investigate the behavior of semi-discrete sampling operators of the Durrmeyer type within the framework of Sobolev-Orlicz spaces. We begin by addressing the problem of modular convergence through a density approach, using the density of compactly supported smooth functions in Sobolev-Orlicz spaces with respect to the modular topology. To this end, we also employ the boundedness of the celebrated Hardy-Littlewood maximal operator in the Orlicz setting. For completeness, we also provide a quantitative analysis to establish sharper estimates based on suitable Orlicz moduli of continuity, in their strong (with respect to the Luxemburg norm) and weak (modular) version. This is obtained through a direct approach based on the Minkowski inequality within the Orlicz framework, involving φ-functions with weaker growth condition at infinity. Consequently, Luxemburg norm and modular convergence theorems in broader functional spaces are derived. Throughout the paper, we discuss several examples of functional spaces, including the Sobolev versions of Zygmund and exponential-type spaces.