Abstract:
This article proves the coincidence of generalized moduli of smoothness of the $k$-s order, defined with the help of
Gegenbauers generalized shift operator, with different and identical shifts and, as a consequence, direct and inverse theorems of
approximation theory by algebraic polynomials are obtained.
Keywords:the best approximation by algebraic polynomials, Gegenbauers generalized shift operator, generalized modulus of smoothness.