Best simultaneous monotone approximants in Orlicz spaces

F. E. LEVIS

La Falda, Córdoba

Conferencia; X Encuentro Nacional de Analistas; 2010

Let M_0 be the class of all real extended mu-measurable functions on [0,1], where mu is the Lebesgue measure.  Let  phi: R_+ ® R_+ be a differentiable and convex function, phi(0)=0, phi(t)>0, t>0. For f Î M_0 let Psi_{phi}(f):=int_0^1 phi(|f(x)|)dmu(x). Several authors studied geometric properties of the Orlicz space L_{phi}[0,1]:={f Î M_0 : Psi_{phi}(lambda f)< ¥ for some lambda > 0 }. Under the Luxemburg norm, L_{phi}[0,1]=:L_{phi}, is a Banach space. It is easy to see that if phi(t)=t^p, 1 0 such that phi(2t)  0 }. Given D Ì L_{phi} and f^j Î L_{phi}, 1