Jackson-type inequalities and widths of functional classes in the Musielak-Orlicz type spaces

TL;DR

This paper establishes precise Jackson-type inequalities and computes various widths for function classes in Musielak-Orlicz spaces, advancing approximation theory in these generalized function spaces.

Contribution

It derives exact Jackson-type inequalities and determines widths of function classes in Musielak-Orlicz spaces, a novel extension in approximation theory.

Findings

Exact Jackson-type inequalities in Musielak-Orlicz spaces

Values of Kolmogorov, Bernstein, linear, and projective widths computed

Results apply to classes of periodic functions with smoothness conditions

Abstract

In the Musielak-Orlicz type spaces ${\mathcal S}_{\bf M}$, exact Jackson-type inequalities are obtained in terms of best approximations of functions and the averaged values of their generalized moduli of smoothness. The values of Kolmogorov, Bernstein, linear, and projective widths in ${\mathcal S}_{\bf M}$ are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.