2019
Том 71
№ 11

On the best $L_1$-approximations of functional classes by splines under restrictions imposed on their derivatives

Abstract

We find the exact asymptotics ($n → ∞$) of the best $L_1$-approximations of classes $W_1^r$ of periodic functions by splines $s ∈ S_{2n, r∼-1}$ ($S_{2n, r∼-1}$ is a set of $2π$-periodic polynomial splines of order $r−1$, defect one, and with nodes at the points $kπ/n,\; k ∈ ℤ$) such that $V_0^{2π} s^{( r-1)} ≤ 1+ɛ_n$, where $\{ɛ_n\}_{n=1}^{ ∞}$ is a decreasing sequence of positive numbers such that $ɛ_n n^2 → ∞$ and $ɛ_n → 0$ as $n → ∞$.

English version (Springer): Ukrainian Mathematical Journal 51 (1999), no. 4, pp 481–491.

Citation Example: Babenko V. F., Parfinovych N. V. On the best $L_1$-approximations of functional classes by splines under restrictions imposed on their derivatives // Ukr. Mat. Zh. - 1999. - 51, № 4. - pp. 435-444.

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