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On Jackson inequality in \(L_ p(\mathbb{T}^ d)\). (Russian. English summary) Zbl 0865.42002

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)
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