The direct and inverse algebraic polynomials approximation theorems in weighted spaces of unbounded functions are proved by using modulus of smoothness. Also, we obtain sharp Jackson ( direct ) inequality of algebraic approximation of unbounded functions in terms modulus of smoothness. In addition, constructive characterization of modulus of smoothness are considered.

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Positive results for spline approximation in weighted space L p,w (X), (1 ≤ p < ∞) are obtained. In particular its show that the degree of best one-sided approximation in terms the k th local modulus of L p,w-continuity and Ditzian-Totic modulus of smoothness for f ∈ L p,w (X).

The purpose of this paper is present a brief survey of know on estimates the rate for best approximation of unbounded functions by suitable trigonometric polynomials of one variable in weighted space , ( ). Moreover we studied concerning the degree of best trigonometric approximation of ( ) with non-integer in , ( ).

The aim of this paper, we introduced some result of best co-approximation specifications of co-proximal and co-chebyshev of unbounded functions in weighted spaces. Also, we discuss some properties of best co-proximal and Co-chebyshev of un bounded functions which gives basic definitions and necessary concepts are convexity, orthogonlity and closed which helps us in the proofs of main results. In addition to that verification, when the function is multiplied by a constant, it maintains its properties.