This article is cited in 3 papers

On the Sharpness of Jackson's Inequality in the Spaces $L_p$ on the Half-Line with Power Weight

V. I. Ivanov

Tula State University

Abstract: In the space $L_p$, $1\leq p<2$, on the half-line with power weight, Jackson's inequality between the value of the best approximation of a function by even entire functions of exponential type and its modulus of continuity defined by means of a generalized shift operator is well known. The question of the sharpness of the inequality remained open. For the constant in Jackson's inequality, we obtain a lower bound, which proves its sharpness.

Keywords: Jackson's inequality, value of the best approximation, the space $L_p$, $1\leq p<2$, entire functions of exponential type, modulus of continuity, generalized shift operator, substochastic matrix, Hoeffding estimate.

UDC: 517.5

Received: 08.05.2015

DOI: 10.4213/mzm10894

 English version: Mathematical Notes, 2015, 98:5, 742–751

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