Trigonometric polynomial approximation, $K$-functionals and generalized moduli of smoothness

Best approximation and approximation by families of linear polynomial operators (FLPO) in the spaces $L_p$, $0<p \le +\infty$, are investigated for periodic functions of an arbitrary number of variables in terms of the generalized modulus of smoothness generated by a periodic generator which, near the origin, is assumed to be close in a certain sense to some homogeneous function of positive order. Direct and inverse theorems (Jackson- and Bernstein-type estimates) are proved; conditions on the generators are obtained under which the approximation error by an FLPO is equivalent to an appropriate modulus of smoothness. These problems are solved by going over from the modulus to an equivalent $K$-functional. The general results obtained are applied to classical objects in the theory of approximation and smoothness. In particular, they are applied to the methods of approximation generated by Fejér, Riesz and Bochner-Riesz kernels, and also to the moduli of smoothness and $K$-functionals corresponding to the conventional, Weyl and Riesz derivatives and to the Laplace operator.
Bibliography: 24 titles.