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Piecewise linear approximation of smooth functions of two variables. (English) Zbl 1285.41010

Summary: Given a piecewise linear (PL) function \(p\) defined on an open subset of \(\mathbb R^n\), one may construct by elementary means a unique polyhedron with multiplicities \(D(p)\) in the cotangent bundle \(\mathbb R^n{\times}\mathbb R^{n\ast} \) representing the graph of the differential of \( p\). Restricting to dimension 2, we show that any smooth function \(f(x,y)\) may be approximated by a sequence \(p_1,p_2,\dots\) of PL functions such that the areas of the \(\mathbb D(p_i)\) are locally dominated by the area of the graph of \(df\) times a universal constant.

MSC:

41A30 Approximation by other special function classes
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