Approximation, characterization, and continuity of multivariate monotonic regression functions

We deal with monotonic regression of multivariate functions [Formula: see text] on a compact rectangular domain in text], where monotonicity is understood generalized sense: as isotonicity some coordinate directions and antitonicity other directions. As usual, the given function that has smallest (weighted) mean-squared distance from text]. establish simple general approach to compute functions: namely, we show can be approximated arbitrarily well — bounds approximation error both text]-norm by grid-constant algorithms. also continuity continuous along an explicit averaging formula for And finally, standard replaced more complex measures which arise, instance, maximum smoothed likelihood estimation. will solution such problems simply