Best one-sided L1-approximation of bivariate functions by sums of univariate ones

Abstract.

Let $f \in C^{1,1} ([-1,1]^2), \partial ^2f/\partial x \partial y \geqq 0$. We characterize the unique best one-sided L ¹-approximant h ^* to f from above (resp. h _* from below) with respect to the subspace B ^1,1 which consists of all bivariate functions which are sums of univariate functions. h ^* resp. h _* are constructed by a Hermite type interpolation on the diagonal $\{ (t,t) : t \in I \}$ resp. the anti-diagonal $\{ (t,-t) : t \in I \}$, where $I := [-1,1]$.