A new concept of smoothness in Orlicz spaces

Abstract

In a 2015 article Cuenya and Ferreyra defined a class of functions in Lp-spaces, denoted by cnp(x). The class cnp(x) contains the class of Lp-differentiability functions, denoted by tnp(x), introduced in a 1961 article by Calderón-Zygmund. A more recent paper by Acinas, Favier and Zó introduced a new class of functions in Orlicz spaces LΦ, called LΦ-differentiable functions in the present article. The class of LΦ-differentiable functions is closely related to the class tnp(x). In this work, we define a class of functions in LΦ, denoted by cnΦ(x). The class cnΦ(x) is more general than the class of LΦ-differentiable functions. We prove the existence of the best local Φ -approximation for functions in cnΦ(x) and study the convexity of the set of cluster points of the set of best Φ -approximations to a function on an interval when their measures tend to zero.