Degree of simultaneous approximation of bivariate functions by Gordon operators

Abstract

We investigate the degree of approximation of bivariate functions on a rectangle by various (discrete) spline-blended operators. Our aim is to give a fuller description than is available in the literature by using mixed moduli of smoothness of higher orders. The crucial tool from the univariate case is a generalization of a theorem of Sharma and Meir on the degree of simultaneous approximation by cubic spline interpolators. The main results for the multivariate case are two theorems expressing certain permanence principles, which explain how the Boolean sums and certain (discrete) blending operators inherit quantitative properties from their univariate building blocks. Various historical remarks and numerous references are included in order to draw the reader's attention to the somewhat diverse history of the subject.