2019
Том 71
№ 6

# Pleshakov M. G.

Articles: 2
Brief Communications (Russian)

### Second Jackson Inequality in a Sign-Preserving Approximation of Periodic Functions

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 123-128

We consider a 2π-periodic function f continuous on $\mathbb{R}$ and changing its sign at 2s points y i ∈ [−π, π). For this function, we prove the existence of a trigonometric polynomial T n of degree ≤n that changes its sign at the same points y i and is such that the deviation | f(x) − T n(x) | satisfies the second Jackson inequality.

Article (Russian)

### Sign-Preserving Approximation of Periodic Functions

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1087-1098

We prove the Jackson theorem for a zero-preserving approximation of periodic functions (i.e., in the case where the approximating polynomial has the same zeros y i) and for a sign-preserving approximation [i.e., in the case where the approximating polynomial is of the same sign as a function f on each interval (y i, y i − 1)]. Here, y i are the points obtained from the initial points −π ≤ y 2s y 2s−1 <...< y1 < π using the equality yi = yi + 2s + 2π furthermore, these points are zeros of a 2π-periodic continuous function f.