Geodesic
Approximation by Piecewise Constant Functions on a Square
Matematičeskie zametki, Tome 75 (2004) no. 4, pp. 592-602

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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. 
@article{MZM_2004_75_4_a7,
     author = {A. S. Kochurov},
     title = {Approximation by {Piecewise} {Constant} {Functions} on a {Square}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {592--602},
     publisher = {mathdoc},
     volume = {75},
     number = {4},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_75_4_a7/}
}
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A. S. Kochurov. Approximation by Piecewise Constant Functions on a Square. Matematičeskie zametki, Tome 75 (2004) no. 4, pp. 592-602. http://geodesic.mathdoc.fr/item/MZM_2004_75_4_a7/