The inverse problem of approximation by algebraic polynomials on manifolds (Q1377932)

Let \(\mathbb{R}^n\) be the \(n\)-dimensional Euclidean space end \(\Gamma= \Gamma_m\), \(1\leq m\leq n\) be an \(m\)-dimensional manifold of this space. We can investigate the problem of approximation of a function \(f(x)\), \(x= (x_1,\dots, x_n)\in \Gamma\) be the algebraic polynomials \[ P_N(x)= \sum_{| x|< N} a_k x^k, \] \(k= (k_1,\dots, k_n)\), \(| k|= \sum^n_{j= 1} k_j\), \(x^k= x^{k_1}_1\cdots x^{k_n}_n\), \(k_j\)-natural numbers. In the papers of the the author [East J. Approx. No. 1, 1-24 (1995; Zbl 0852.41018); Tr. Mat. Inst. Steklova 204, 201-225 (1993; Zbl 0845.41016); Dokl. Akad. Nauk SSSR 317, No. 1, 44-46 (1991; Zbl 0763.41006)] it was showed, that every function \(f\in H^r_p(\Gamma)\) can be approximated by the polynomials \(P_N\) on \(\Gamma\) with the estimates \[ \| f- P_N\|_{L_p(\Gamma)}\leq c N^{-r}\| f\|_{H^r_p(\Gamma)},\tag{1} \] \[ \| P_N\|_{L_p(\Delta)}\leq c\| f\|_{H^r_p(\Gamma)},\quad \Gamma\in C^k,\tag{2} \] where \(c\) is an independent constant, \(\Gamma\) is a closed bounded manifold of smoothness \(k>r\) and \(\Delta\) is a rectangle, \(\Gamma\subset \Delta\subset \mathbb{R}^n\). In this paper, an inverse approximation problem is investigated: to determine some conditions, that should be imposed on \(\Gamma\), in order that the following implication holds (1) an d(2) \(\Rightarrow f\in H^r_p(\Gamma)\). The author says, that the last statements is valid in any case if, for \(\Gamma\), an equality of Bernstein type for polynomials \(P_N\) holds. In this case, the conditions (1) and (2) imply, that a stronger inequality actually holds for \(P_N\), \(\Gamma\in C^k\), \(k>r\), \[ \| P_N\|_{H^r_p(\Gamma)}\leq c\| f\|_{H^r_p(\Gamma)},\quad N>1,\tag{\(2'\)} \] and that (1) and (2) imply in a trivial way that \(f\in H^r_p(\Gamma)\). The inverse approximation problem is investigated under the assumption that \(\Gamma\) is defined parametrically by means of trigonometric polynomials.