Moduli of continuity defined by zero continuation of functions andK-functionals with restrictions

Abstract

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 ^r _p,U is a subspace of the Sobolev spaceW ^r _p [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 =τ _s forj ≠s, 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.