About the relation between the best approximations in mixed norms

Abstract

The first time the ratio between the best approximations of polynomials with harmonics from hyperbolic crosses, corresponding to a given mixed derivative in the isotropic space was proved by V.N.Temlyakov ([1], 2.3 Theorem) The main theorem of this article is devoted to the generalization of the relation obtained by V.N.Temlyakov, for the anisotropic case. The main problem generating some difficulties here is finding links between p = (p_1, p_2, ..., p_d), q = (q_1, q_2, ..., q_d), r = (r_1, r_2, ..., r_d) etc. parameters along with proof of the inequality of Bernstein, which plays an important role in the anisotropic space. Due to the complexities of these issues, in 1991, E.Aidos got the correlation between the best approximations in different mixed norms for cases where the above parameters satisfy certain specific conditions. In this article for relations between best approximations in different mixed norms, these conditions are removed, i.e. obtained for the general case when the basic parameters are of the form 1 < p_i < q-i < ∞, i = 1, ..., d, r = (r_1, ..., r_d), min r_i ≥ 0. In the article the relation between the best approximations in the norms L_p(π_d) and L_q(π_d) is expressed in terms of trigonometric polynomials whose harmonics lie in hyperbolic crosses, corresponding to a given mixed derivative. From the inequality, pointed in the theorem, we can obtain attachments of types E_p,d,Q(λ) ⊂ L_q^(r)(πd), E_p,d,Q(λ) ⊂ E_q,d,Q^(r)(μ).