Pasko A. One-sided approximations to the functions by algebraic polynomials

The object of the research are classes of the functions defined on the segment, assigned with restriction on uniform or average norm or modulus of continuity of the r-th derivative. The purpose is studying of the best one-sided approximation by the algebraic polynomials on the segment. Methods of research are modern methods of the functional theory, mathematical analyses and approximation theory. The asymptotic estimation of the best one-sided approximations by algebraic polynomials in mean to the classes assigned with the restrictions on the average norm of the r-th derivative was found. Asymptotic estimation of the best one-sided weighted approximations to these classes in mean, with weight function satisfying certain conditions, was established. The remainder in V.G. Doronin and A.A. Ligun estimation of the best one-sided approximations with considering the location of the point on the segment was improved. Analogue for one-sided case of V.P. Motornyi asymptotic estimation of the best approximationwith considering the location of the point on the segment was received. All results are new, they are published in the advanced scientific magazines and may be useful for further development of approximation theory.