Exact constants of approximation for differentiable periodic functions

For all odd r we construct a linear operator B_r,r(f) which maps the set of 2-periodic functionsf(t) X^(r) (X^(r)=C^(r) or L₁ ^(r)) into a set of trigonometric polynomials of order not higher than n-1 such that where X is the C or L₁ metric, E_n(f)_X and (f, )_X are the best approximation by means of trigonometric polynomials of order not higher than n-1 and the modulus of continuity of the functionf in the X metric, respectively; K_r are the known Favard constants.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 21–30, July, 1973.In conclusion, the author wishes to express his deep gratitude to N. P. Korneichuk under whose guidance this paper was written.