Best Approximation of Unbounded Functions in L_(p,α) (X)

Abstract

This work is devoted to approximate unbounded functions in weighted space and deal with what is called approximation by algebraic and trigonometric polynomials in the approximation theory. The direct and converse algebraic polynomials approximation theorems of unbounded functions in weighted spaces are proved by using modulus of smoothness. Approximation by polynomials in the space L_(p,α) (X) is concerned mainly with algebraic polynomial approximation of unbounded function in weighted space in terms convolution f*g, where f and g belongs to weighted space. In addition, we obtain the best approximate the function to fractional derivatives in L_(p,α) (X). We investigate some improvement to the theorems of direct and converse trigonometric approximation of unbounded functions in weighted spaces to obtain strong inverse inequalities for some methods of approximating functions using the modulus of smoothness. Furthermore, we study the approximate properties of functions by means of trigonometric polynomials in weighted spaces. Relationships between modulus of smoothness of function derivatives and those of the jobs themselves are introduced. In the weighted spaces we also proved of theorems about the relationship between the derivatives of the polynomials for the best approximation of the functions