2019
Том 71
№ 11

All Issues

Popov P. A.

Articles: 3
Brief Communications (Russian)

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

Pleshakov M. G., Popov P. A.

↓ Abstract   |   Full text (.pdf)

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

Pleshakov M. G., Popov P. A.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

An Analog of the Jackson Inequality for Coconvex Approximation of Periodic Functions

Popov P. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 919-928

We prove an analog of the Jackson inequality for a coconvex approximation of continuous periodic functions with the second modulus of continuity and a constant that depends on the location of the points at which a function changes its convexity.