Approximation properties of periodic multivariate quasi-interpolation operators

Abstract

We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions ${\tilde{\phi }}_j$ and trigonometric polynomials ${\phi }_j$. The class of such operators includes classical interpolation polynomials (${\tilde{\phi }}_j$ is the Dirac delta function), Kantorovich-type operators (${\tilde{\phi }}_j$ is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on ${\tilde{\phi }}_j$ and ${\phi }_j$, 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.