Pointwise Estimates for Finding the Error of Best Approximation by Spline, Positive Algebraic Polynomials and Copositive

Abstract

     The first step in this research is to find some of the necessary estimations in approximation by using certain algebraic polynomials, as well as we use certain specific points in approximation. There are many estimations that help to find the best approximation using algebraic polynomials and geometric polynomials. Throughout this research, we deal with some of these estimations to estimate the best approximation error using algebraic polynomials where the basic estimations in approximation are discussed and proven using algebraic polynomials that are discussed and proven using algebraic polynomials that are specified by the following points and  if   as well as if   .

  For the second step of the work, the estimations in the first step are used to find and estimate the error for the best approximation of the weighted function . This is done through the use of an algebraic polynomial whose degree at most is where the sign of the algebraic polynomial is positive.

   Further, the error is also found and estimated for the best approximation of the restricted function    using the restricted algebraic polynomial , which is copositive with the function   in the quasi weighted normed space.

    In addition, we deal with the created estimations to estimate the error of the best approximation of the function   by using pieces of algebraic polynomials that are of the highest degree .These pieces of algebraic polynomials are connected to each other, so they have formed a spline of the highest degree  whose knots are considered the contact areas of the algebraic polynomials.