turkmath.org

In the paper we consider some problems of the theory of approximation of functions on interval $[0,\infty)$ in the metric of $L_{p,\lambda}$ with weight $\sinh^{2\lambda} x$ using generalized Gegenbauer shifts. We prove analogues of direct Jackon theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Gegenbauer shifts. We establish the equivalence of the modulus of smoothness and $K$-functional, defined in terms of the space of Sobolev type corresponding to the Gegenbauer differential operator. The research of E. Ibrahimov was supported by the grant of Presidium of Azerbaijan National Academy of Science 2015.