Best weighted approximation and its degree for (co) convex and unconstrained polynomials

Approximation theory is a branch of analysis and applied mathematics that approximates a function f which is defined on a finite interval [a,b] with preserving certain intrinsic shape properties and known as concavities of the functions. The study of (co)convex and unconstrained polynomial (COCUNP) are the important properties in finding the best approximation of a function. Numerous studies have been conducted using COCUNP approximation to construct a degree of the best approximation, but it depended on the Ditzian-Totik modulus of smoothness (DTMS) of f and also limited to a uniform norm space. Moreover, this approach could not be extended to the best weighted approximation of functions in general abstract spaces. Thus, to overcome this shortcoming, this study established a new symmetric difference using properties of Lebesgue Stieltjes integral-i which is called generalization of symmetric difference. Then, the best weighted approximation known as weighted DTMS is developed by taking the supremum of generalization of symmetric difference. In addition, new degrees of best (co)convex and unconstrained polynomial approximation are constructed using the proposed weighted DTMS. The result shows that the new approach is able to extend the best weighted approximation of functions in general abstract spaces. In addition, the weighted approximation by (co)convex polynomial provided a better accuracy compared to unconstrained polynomial. In conclusion, this study has successfully established the best weighted approximation and improved its degrees for (co)convex polynomial and COCUNP.