Derivation of new Degrees for Best (Co)convex and Unconstrained Polynomial Approximation in ${\mathbb{L}}_p^{\alpha ,\beta }$ space: I

The purpose of this work is to present the derivation and an estimate of the degrees of the best approximation based on convex, coconvex and unconstrained polynomials, and discuss some applications. We simplify the term convex and coconvex polynomial as (co)convex polynomial herein.