Rozhdestvenskij, A. V. On Jackson inequality in \(L_ p(\mathbb{T}^ d)\). (Russian. English summary) Zbl 0865.42002 Fundam. Prikl. Mat. 1, No. 3, 753-766 (1995). Summary: The author proved some necessary and sufficient conditions on a finite set of \(d\)-dimensional vectors \(\{\alpha_l\}\) for the Jackson-Yudin inequality to be satisfied for the approximation of periodic functions \(f\) by trigonometric polynomials: \[ E_{n-1}(f)_q\leq A\cdot n^{-r+(d/p-d/q)_+}\cdot\max_l |\Delta^m_{2\pi\alpha_l/n}f^{(r)}|_p, \] where \(A>0\) is independent of \(f\) and \(n\).A criterion for solvability of the homological equation \[ f(x)-{1\over(2\pi)^d}\int f(t)dt=\varphi(x+2\pi\alpha)-\varphi(x)\qquad\text{a.e. }x \] on the sets of functions \(\{f:f^{(r)}\in L_p(\mathbb{T}^d)\}\) is obtained. MSC: 42A10 Trigonometric approximation 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) Keywords:Jackson-Yudin inequality; approximation of periodic functions; trigonometric polynomials PDFBibTeX XMLCite \textit{A. V. Rozhdestvenskij}, Fundam. Prikl. Mat. 1, No. 3, 753--766 (1995; Zbl 0865.42002)