Error Estimation for polynomial Approximations in L_(p,w^* ) space

Abstract

In this research ,we conduct an analytical study to estimate the approximate error of  functions using algebraic polynomials in the normed linear space    ,  .We first address the limitations of the weighted integral operator and  then establish a mathematical relationship between the norm of the weight functions    and the norm of continuous functions  on the uniform partition of  the interval .We found a relationship between higher-order smoothness and the approximation      error ,and we also determined approximation constants that depend on the properties of the normed linear space .Furthermore, we established the interpolation of functions  on the interval  where , to minimize the difference between the function and its derivatives and the nearest algebraic polynomials in representation. We also demonstrated that the uniform  normed space on the interval  can be controlled by the normed linear space .