On some extremal problems of approximation theory of functions on the real axis I.

A number of extremal problems of approximation theory of functions have been solved on the real line $$ \mathbb{R} $$ . Exact constants in the Jackson-type inequalities in the space L( $$ \mathbb{R} $$) are calculated. The exact values of average ν-widths are obtained for the classes of functions from L( $$ \mathbb{R} $$) defined by averaged moduli of continuity of the k-th order, as well as for the classes of functions defined by K-functionals. The quite complete analysis of the final results related to the solution of extremal problems of approximation theory in the periodic case and for the whole real axis is carried out in the chronological order. [ABSTRACT FROM AUTHOR]