Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions - Radzievskii, G. V. (Ukrains’kyi Matematychnyi Zhurnal, 1996)
We consider the followingK-functional: K(δ ,f)_p : = _g W_p U^r { \| f - g \|_L_p + δ _j = 0^r \| g^(j) \|_L_p }, δ 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 \(_0^1 g^(l_j ) (τ ) dσ _j (τ ) = 0, j = 1, ... , n \) . 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 \(f(t) = f(t)\) , 0≤t≤1, let \(f(t) = 0\) ,t0 does not depend on δ>0 and ƒ ∈L p . We also establish some other estimates for the consideredK-functional.