Approximation by Piecewise Constant Functions on a Square

In this paper we consider several algorithms for approximating functions defined on the unit square ${\mathbf I}= [0,1]^2$ and ranging in $\mathbb{R}^2$. We use functions of zeroth-order Lagrange spline type as the approximation apparatus. They differ from the standard Lagrange splines on the plane by the rule for choosing grid lines according to which the spline is constructed; namely, a set of one-dimensional splines is used instead of a family of parallel lines determining the interpolation nodes.