(FUNDAMENTAL AND APPLIED MATHEMATICS)

1995, VOLUME 1, NUMBER 3, PAGES 753-766

## On Jackson inequality in $L_p(\ttt^d)$

A.V.Rozhdestvensky

The author proved some necessary and sufficient conditions on a finite set of d--dimensional vectors $\{\alpha_l\}$, when Jackson--Youdin inequality for the approximation of periodic function f by trigonometric polynomials:

E_{n-1}(f)_q \leq A\cdot n^{-r +(d/p-d/q)_+} \cdot \max_{l} \| \Delta_{2\pi\alpha_l/n}^m f^{(r)} \|_p,
where A > 0 is independent of f and n, holds.

A criterion of solvability of the homological equation

f(x) - \frac{1}{(2\pi)^d} \int f(t) dt = \varphi (x+2\pi\alpha) - \varphi(x) \qquad a.e. x
on the sets of functions $\{f\colon f^{(r)} \in L_p(\mathbb{T}^d)\}$ is obtained.

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