Regular multivariate sampling and approximation in Lp shift-invariant spaces

In this paper some results about regular multivariate generalized sampling in the Lp setting (1⩽p⩽∞) are proven. Thus, stable multivariate regular sampling formulas are derived for Lp shift-invariant spaces , i.e., the Lp-closure of the linear span of the shifts of a finite set Φ of generators. These sampling formulas include regular samples of the function, its derivatives and/or some filtered versions of the function itself taken at a lattice of Rd. Approximation schemes using these generalized sampling formulas are also included.