Abstract
Given a continuous function f defined on the unit cube ofR n and a convex function φt, φt(0)=0, φt(x)>0, for x>0, we prove that the set of best Lφt-approximations by monotone functions has exactly one element ft, which is also a continuous function. Moreover if the family of convex functions {φt}t>0 converges uniformly on compact sets to a function φ0, then the best approximation ft→f0 uniformly, as t→0, where f0 is the best approximation of f within the Orlicz space L0. The best approximations {ft} are obtained as well as minimizing integrals or the Luxemburg norm.
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