Approximation properties of periodic multivariate quasi-interpolation operators
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We study approximation properties of the general multivariate periodic quasi-interpolation operator Q_j(f,φ_j,φ_j), which is generated by the distribution/function φ_j and some trigonometric polynomial 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 estimates for the L_p-error of approximation by operators Q_j(f,φ_j,φ_j) in terms of the best and best one-sided approximation, classical and fractional moduli of smoothness, K-functionals, and other terms.
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