Some estimates for octonion transform

Abstract

This paper extends classical approximation theory by establishing Bernstein’s inequality and Jackson’s direct and inverse theorems for the Octonion transform, focusing on functions with a bounded spectrum. Jackson’s inequality bounds the best polynomial approximation of a function based on its smoothness, while Bernstein’s inequality relates the maximum modulus of a polynomial and its derivative on the unit disk. The paper adapts these concepts to the context of octonions.