On the Relationship Between Smoothness and Best Approximation in Weighted Function Space

Abstract

This work investigates the relationship between the smoothness measure and the function norm in the space  , where modified symmetric difference based on the Ditzian-Totik function  are employed to assess the behavior of functions within centered subintervals. The focus is on the weighted smoothness measure of order  used to derive both local and global estimates of the function. The weighted smoothness measure can be bound in terms of a series involving  , which represents the optimal approximation of the function by polynomials. The following estimate incorporates the effects of the weight, partial smoothness, and derivative behavior into a precise quantitative expression linking approximation properties with smoothness analysis. The results contribute to a deeper understanding of the interplay between function smoothness and behavior in approximation function spaces, and they open pathways to accurate numerical applications.