(untitled) (JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences, 2020)
Moduli of smoothness are intended for mathematicians working in approximation theory, numerical analysis and real analysis. Measuring the smoothness of a function by differentiability is two grade for many purposes in approximation theory. More subtle measurement are provided by moduli of smoothness. Many versions of moduli of smoothness and K-functionals introduced by many authors. In this work we choose two of these moduli and prove that they are equivalent themselves once and with a version of K-functional twice, under certain conditions. As an application of our work we introduce a version of Jackson theorem for the approximation by neural networks.