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 s...
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| Date: | 1996 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1996
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5208 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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 =τ s forj ≠s, thenl j ≠l s . Let \(\hat f(t) = f(t)\) , 0≤t≤1, let \(\hat f(t) = 0\) ,t0 does not depend on δ>0 and ƒ ∈L p . We also establish some other estimates for the consideredK-functional. |
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