A new concept of smoothness in Orlicz spaces

In a 2015 article Cuenya and Ferreyra defined a class of functions in $$L^p$$ -spaces, denoted by $$c_n^p(x)$$ . The class $$c_n^p(x)$$ contains the class of $$L^p$$ -differentiability functions, denoted by $$t_n^p(x)$$ , introduced in a 1961 article by Calderon-Zygmund. A more recent paper by Acinas, Favier and Zo introduced a new class of functions in Orlicz spaces $$L^\Phi$$ , called $$L^\Phi$$ -differentiable functions in the present article. The class of $$L^\Phi$$ -differentiable functions is closely related to the class $$t_n^p(x)$$ . In this work, we define a class of functions in $$L^\Phi$$ , denoted by $$c_n^{\Phi }(x)$$ . The class $$c_n^{\Phi }(x)$$ is more general than the class of $$L^{\varPhi}$$ -differentiable functions. We prove the existence of the best local $$\Phi$$ -approximation for functions in $$c_n^{\varPhi }(x)$$ and study the convexity of the set of cluster points of the set of best $$\Phi$$ -approximations to a function on an interval when their measures tend to zero.