Approximation from shift-invariant spaces by generalized sampling formulas

In this paper we investigate approximation from shift-invariant spaces by using generalized sampling formulas. These sampling formulas involve samples of filtered versions of the function itself. The considered systems include averaging samplers, and classical sampling of the function and its derivatives. Under appropriate hypotheses on the generator φ of the shift-invariant space and on the involved systems, we derive stable generalized sampling formulas in a shift-invariant subspace of Lp(Rn). From these generalized sampling formulas we construct approximation schemes valid for smooth functions. The approximation order depends both on the order for which the Strang–Fix conditions are satisfied by φ, and on the largest order of the derivatives appearing in the systems if any.