Regular multivariate sampling and approximation in shift-invariant spaces

In this paper some results about regular multivariate generalized sampling in the L p setting ( 1 ⩽ p ⩽ ∞ ) are proven. Thus, stable multivariate regular sampling formulas are derived for L p shift-invariant spaces V Φ p , i.e., the L p -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 R d . Approximation schemes using these generalized sampling formulas are also included.