Abstract
Abstract For 2π-periodic functions from Lp (where 1 < p < ∞) we prove an estimate of approximation by Euler means in Lp metric generalizing a result of L. Rempuska and K. Tomczak. Furthermore, we show that this estimate is sharp in a certain sense. We study the uniform approximation of functions by Euler means in terms of their best approximations in p-variational metric and also prove the sharpness of this estimate under some conditions. Similar problems are treated for conjugate functions.
Highlights
IntroductionFor 1 < p < ∞ let us introduce the space Vp of all 2π-periodic bounded functions with the property
Let 1 < p < ∞, f be a 2π-periodic real measurable bounded function, ξ = {x0 < x1 < . . . < xn = x0 + 2π} be a partition of a period and p ξ (f ) := (︂ n ∑︀ |f (xi) − f)︂1/p (xi−1)|p i=1For 1 < p < ∞ we define ω1−1/p(f, δ) to be ω1−1/ p (f, δ) =
For 2π-periodic functions from Lp we prove an estimate of approximation by Euler means in Lp metric generalizing a result of L
Summary
For 1 < p < ∞ let us introduce the space Vp of all 2π-periodic bounded functions with the property. The space Vp of functions of bounded p-variation was introduced in the case p = 2 by Wiener [2] while the space Cp of p-absolutely continuous functions in another but equivalent form was considered by Young [3] (see the paper of Love [4]) Both Vp and Cp are Banach spaces with respect to ‖ · ‖Vp. For 1 ≤ p < ∞ let Lp be the space of 2π-periodic measurable functions with finite norm. We obtain the degree of approximation of bounded p-variation functions by Euler means in uniform norm and show its sharpness under some additional conditions.
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