Estimates of the best approximations of periodic functions by trigonometric polynomials in terms of averaged differences and the multidimensional Jackson's theorem

In the first section the best approximations of periodic functions of one real variable by trigonometric polynomials are studied. Estimates of these approximations in terms of averaged differences are obtained. A multidimensional generalization of these estimates is presented in the second section. As a consequence. The multidimensional Jackson's theorem is proved.