Bozhukha L. N.
On a Jackson-Type Inequality in the Approximation of a Function by Linear Summation Methods in the Space $L_2$
Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 537-545
We prove a statement on exact inequalities between the deviations of functions from their linear methods (in the metric of $L_2$) with multipliers defined by a continuous function and majorants determined as the scalar product of the squared modulus of continuity (of order r) in $L_2$ for the lth derivative of the function and a certain weight function θ. We obtain several corollaries of the general theorem.
Inequalities of the Jackson Type in the Approximation of Periodic Functions by Fejér, Rogosinski, and Korovkin Polynomials
Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1596-1602
We consider inequalities of the Jackson type in the case of approximation of periodic functions by linear means of their Fourier series in the space L 2. In solving this problem, we choose the integral of the square of the modulus of continuity as a majorant of the square of the deviation. We establish that the constants for the Fejér and Rogosinski polynomials coincide with the constant of the best approximation, whereas the constant for the Korovkin polynomials is greater than the constant of the best approximation.