One – Sided Approximation in Lp(X)

The aim of this research to study the approximation of functions in the space- 𝐿𝑝 by the “algebraic polynomial” in terms of the” average modulus” of the k-order also, we will estimate the degree of the (O-S- A), (means one – sided approximation) in term of averaged modulus where all the results which number is eleven we need to prove the main theorem that (the degree of best (O-S- A) of 𝑓 by trigonometric polynomials of order 𝑛 in 𝐿 𝑝 (𝑋 ), (𝐸̃𝑛 (𝑓)𝑝) ) is less than or equal to (Averaged modulus of smoothness of 𝑓 of order- 𝑘, (𝜏𝑘 (𝑓 , 1 𝑛 ) 𝑝 ) ) have been proven, It has also been proven the converse theorem for the main theorem in this research.