Exact estimation of an approximation of some classes of differentiable functions by convolution operators

We obtained explicit formulas for the calculation of a value of the approximation of the classes W ^r,β _p,n by convolution operators of a special form. Here, ( beta in mathbb{Z} ) , r > 0, ( n in mathbb{N} ), and p = 1 or p = ∞. As partial cases, we deduced explicit formulas for a value of the approximation of the indicated classes by generalized Abel–Poisson means, biharmonic Poisson operators, and Riesz and Cesáro means. In some cases, we constructed the asymptotic expansions in a parameter for a value of the approximation of the indicated classes. For natural r, some results were obtained by Nikol’skii, Nagy, Timan, Telyakovskii, Baskakov, Falaleev, Kharkevych, and other mathematicians.