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 φj. The class such operators includes classical interpolation (φ˜j is the Dirac delta function), Kantorovich-type a characteristic scaling expansions associated with wavelet constructions, others. Under different compatibility conditions on φj, we obtain upper lower bound estimates for Lp-error in terms best one-sided approximation, fractional moduli smoothness, K-functionals, other terms.