Approximation properties of periodic multivariate quasi-interpolation operators

We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions φ˜_j and trigonometric polynomials φ_j. The class of such operators includes classical interpolation polynomials (φ˜_j is the Dirac delta function), Kantorovich-type operators (φ˜_j is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on φ˜_j and φ_j, we obtain upper and lower bound estimates for the L_p-error of approximation by quasi-interpolation operators in terms of the best and best one-sided approximation, classical and fractional moduli of smoothness, K-functionals, and other terms.