Approximation in <em>S</em>-normed Orlicz Spaces

Orlicz spaces generalize conventional Lebesgue spaces, providing a more flexible framework for analyzing functions. They are essential to functional analysis and related fields, particularly in approximation theory. This paper defines an Orlicz normed space equipped with an S-norm, where S is an outer function that satisfies certain conditions. Introducing norms via the outer function S is a general method that encompasses many cases of norms in Orlicz spaces. The relationship between S-norm and other norms on Orlicz spaces is established through several equivalences. Moreover, moduli of smoothness are of great interest in approximation theory, as they provide criteria for assessing the accuracy of approximations. Accordingly, the S-norm was used to define the S-modulus of smoothness of fractional order. Fundamental mathematical properties were also presented to support the proofs of the main approximation theorems. Furthermore, the modulus was employed to prove direct and inverse theorems by studying the degree of best approximation. This demonstrates that the upper and lower bounds of the degree of approximation approach zero as the S-modulus of smoothness tends to zero.