f23f59b1-17da-4e24-8ad1-f1e61009ab8020250127065137909wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40511220241220242310.37394/23206.2024.23https://wseas.com/journals/mathematics/2024.phpAli HusseinZaboonDepartment of Education Supervision Iraqi Ministry of Education, University of Mustansiriyah, Baghdad, IRAQThe 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.123120241231202494094797https://wseas.com/journals/mathematics/2024/b945106-2048.pdf10.37394/23206.2024.23.97https://wseas.com/journals/mathematics/2024/b945106-2048.pdf10.1016/0021-9045(85)90050-4G. Nurnberger, Unicity in one-sided approximations 𝐿 1 − approximations and quadrature formulae, Journal of Approximation Theory, Vol.45, Issue 3, pp.271-279, 1985, https://doi.org/10.1016/0021-9045(85)90050- 4. 10.1007/bf01058514V.F. Babenko and V.N. Glushko, On the uniqueness of elements of the best approximation and the best one -sided approximation in 𝐿 1 −space, Ukrainian Mathematical Journal, Vo.46, No.5, pp.503- 513, 1994, https://doi.org/10.1007/BF01058514. 10.1216/rmj-2011-41-5-1725C. Yang, one-sided 𝐿 𝑝 −norm and best approximation in one- sided𝐿 𝑝 −norm, The Rocky Mountain Journal of Mathematical, Vo.41, Issue 5, pp. 1725-1740, 2011, DOI: 10.1216/RMJ-2011-41-5-1725. 10.1007/s11253-017-1359-xV. V. Sedunova, best one-sided approximations of the class of differentiable functions by algebraic polynomials in the mean, Ukrainian MTh. Journal, Vol. 69, pp.255-268, 2017, https://doi.org/10.1007/s11253-017-1359-x. 10.1016/j.jat.2015.05.001J. Bustamante, Reinaldo M. and Jose M. Quesada, Quasi orthogonal Jacobi polynomials and best one-sided approximation to step functions, Journal of Approximation Theory, Vol.198, Issue 10, pp.10-23, 2015, https://doi.org/10.1016/j.jat.2015.05.001. 10.18052/www.scipress.com/ijarm.6.42Alexander N. Shchitov, Best One-Sided Approximation of Some Classes of Functions of Several Variables by Haar Polynomials, International Journal of Advanced Research in Mathematics, Vol. 6, ISSN: 2297-6213, pp. 42-50, 2016, [Online]. https://www.researchgate.net/publication/308 720960 (Accessed Date: December 2, 2024). A. A. Auad and Mousa M. Khrajan, Direct Estimation for One-Sided Approximation by Polynomial Operators, MJPS, Vol. (3), No. (2), pp.54-59. Babenko A. G., One-sided approximation in L of the characteristic function of an interval by trigonometric polynomials, Trudy institnte Mathematica mekhanki, Uro RAN m, Vol.18 No.1, pp.39-52, [Online]. https://elar.urfu.ru/bitstream/10995/27301/1/s copus-2013-0464.pdf (Accessed Date: December 2, 2024). Blagovest Sendov and Vasil A. Popov, The Averaged Moduli of Smoothness, Bulgarian Academy of Sciences University of St Andrews, pp.163-171, 1988. 10.1088/1742-6596/2322/1/012018Ali Hussein Zaboon, Approximation of Functions in 𝐿𝑝,𝒞𝑛 (𝕏)-space, Journal of Physics: Conference Series 2322, 012018, Issue 1, pp. 1-5, [Online]. https://www.researchgate.net/publication/363 101861 (Accessed Date: December 2, 2024).