Quasi-projection operators with applications to differential-difference expansions

A large class of multivariate quasi-projection operators is studied. These operators are sampling-type expansions Sigma(k is an element of Zd) c(k) (f)psi(A(j). - k), where A is a matrix and the coefficients c(k)(f) are associated with a tempered distribution (psi) over tilde. Error estimates in L-p-norm, 2 <= p <infinity, are obtained under the so-called weak compatibility conditions for <(psi)over tilde> and psi, and the Strang-Fix conditions for psi. Approximation order depends on the smoothness of f and on the order of the compatibility and the Strang-Fix conditions. A number of examples aimed at engineering applications are provided. A special attention is payed to the quasi-projection operators associated with differential-difference expansions. Applications to solving some differential-difference equations are presented. (C) 2019 Elsevier Inc. All rights reserved.