Best one-sided L-1-approximation of bivariate functions by sums of univariate ones

Let f is an element of C-1,C-1([-1,1](2)), partial derivative(2)f/partial d erivative x partial derivative y greater than or equal to 0. We characteriz e the unique best one-sided L-1-approximant h* to f from above (resp. h(*) from below) with respect to the subspace B-1,B-1 which consists of all biva riate functions which are sums of univariate functions. h* resp. h(*) are c onstructed by a Hermite type interpolation on the diagonal ((t.t) : t is an element of I) resp. the anti-diagonal ((t, -t) : t is an element of I), wh ere I:= [-1,1].