K –Mixed Modulus of Smoothness

Abstract

The modulus of smoothness is essential for modern analysis and its applications. Is a versatile tool in approximation theory that helps in understanding the properties of approximation methods, characterizing function spaces, and analyzing the convergence and accuracy of numerical algorithms. It helps in determining the optimal number of terms or the optimal choice of basic functions to achieve the desired level of accuracy in approximation of a function and rate of convergence. The smoothness modulus has many applications, including its applications in numerical analysis, particularly in the analysis of numerical methods for solving differential equations, optimization problems, and integral equations. Many papers introduced the ordinary modulus of smoothness with one variable. However, few researchers have tried to approximate functions with multiple variables and mixed modulus. This paper tries to fill in that gap introducing a new  -mixed modulus of smoothness for measurable functions  with d –variables. It does this by using a new  –mixed difference and proving some of its approximation properties, like linearity and monotonicity, for the  -mixed modulus of smoothness of functions belonging to the space  using the vector of numbers , . Also, study the approximation of bounded functions with  -mixed difference and its direct relationship with mixed smoothness.

Received 28/09/2023

Revised 01/03/2024

Accepted 03/03/2024

Published Online First 20/07/2024