2018
Том 70
№ 8

All Issues

Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order

Radzievskaya E. I., Radzievskii G. V.

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Abstract

Let U j be a finite system of functionals of the form \(U_j (g):= \int _0^1 g^(k_j) ( \tau ) d \sigma _j ( \tau )+ \sum_{l < k_j} c_{j,l} g^(l) (0)\) , and let \(W_{p,U}^r\) be the subspace of the Sobolev space \(W_p^r [0;1]\) , 1 ≤ p ≤ +∞, that consists only of functions g such that U j(g) = 0 for k j < r. It is assumed that there exists at least one jump τ j for every function σ j , and if τ j = τ s for js, then k jk s. For the K-functional $$K(\delta, f; L_p ,W_{p,U}^r) := \inf \limits_{g \in W_{p,U}^r} {|| f-g ||_p + \delta (|| g ||_p + || g^(r) ||_p)},$$ we establish the inequality \(K(\delta^n , f;L_p ,W_{p,U}^r) \leqslant cK(\delta^r ,f; L_p ,W_{p,U}^r)\) , where the constant c > 0 does not depend on δ ε (0; 1], the functions f belong to L p, and r = 1, ¨, n. On the basis of this inequality, we also obtain estimates for the K-functional in terms of the modulus of smoothness of a function f.

English version (Springer): Ukrainian Mathematical Journal 55 (2003), no. 11, pp 1841-1852.

Citation Example: Radzievskaya E. I., Radzievskii G. V. Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order // Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1530-1540.

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