2019
Том 71
№ 11

# Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions

We consider the followingK-functional: $$K(\delta ,f)_p : = \mathop {\sup }\limits_{g \in W_{p U}^r } \left\{ {\left\| {f - g} \right\|_{L_p } + \delta \sum\limits_{j = 0}^r {\left\| {g^{(j)} } \right\|_{L_p } } } \right\}, \delta \geqslant 0,$$ where ƒ ∈L p :=L p [0, 1] andW p,U r is a subspace of the Sobolev spaceW p r [0, 1], 1≤p≤∞, which consists of functionsg such that $\int_0^1 {g^{(l_j )} (\tau ) d\sigma _j (\tau ) = 0, j = 1, ... , n}$ . Assume that 0≤l l ≤...≤l n r-1 and there is at least one point τ j of jump for each function σ j , and if τ j s forjs, thenl j l s . Let $\hat f(t) = f(t)$ , 0≤t≤1, let $\hat f(t) = 0$ ,t<0, and let the modulus of continuity of the functionf be given by the equality $$\hat \omega _0^{[l]} (\delta ,f)_p : = \mathop {\sup }\limits_{0 \leqslant h \leqslant \delta } \left\| {\sum\limits_{j = 0}^l {( - 1)^j \left( \begin{gathered} l \hfill \\ j \hfill \\ \end{gathered} \right)\hat f( - hj)} } \right\|_{L_p } , \delta \geqslant 0.$$
We obtain the estimates $K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 ]} (\delta ,f)_p$ and $K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 + 1]} (\delta ^\beta ,f)_p$ , where β=(pl l + 1)/p(l 1 + 1), and the constantc>0 does not depend on δ>0 and ƒ ∈L p . We also establish some other estimates for the consideredK-functional.