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

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 j ≠ s, then k _j ≠ k _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.