Best one-sided \(L^1\)-approximation of bivariate functions by sums of univariate ones (Q1587794)

Let \[ D= \Biggl\{f \in C[-1,1]: { {\delta^2}\over {\delta x\delta y}} f \geq 0 \Biggr\}. \] Let \(B\) be the functions on \([-1,1]\) that are the sums of continuously differentiable functions of one variable. For each \(f\in D\) this paper proves the existence and characterization of a unique one-sided best approximation from \(B\). Their striking result is the following: Theorem: Let \(f\in D\). There is a unique \(b\in B\) such that \(b\geq f\) and if \(g\in B\) is also such that \(g\geq f\), then \[ \iint|f-b|dx dy \leq \iint|f-g|dx dy. \] Furthermore \(b\) is the unique member of \(B\) such that for all \(-1\leq t \leq 1\) (i) \(b(t,t)=f(t,t),\) and (ii) \(\operatorname {grad} b(t,t)= \operatorname {grad} f(t,t)\).