Geodesic
The Kolmogorov and Stechkin Problems for Classes of Functions Whose Second Derivative Belongs to the Orlicz Space
Matematičeskie zametki, Tome 91 (2012) no. 2, pp. 172-183

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For any $t\in [0,1]$, we obtain the exact value of the modulus of continuity $$ \omega_N(D_t,\delta):=\sup\{|x'(t)|:\|x\|_{L_{\infty}[0,1]}\le \delta,\, \|x''\|_{L_{N}^*[0,1]}\le 1\}, $$ where $L_N^*$ is the dual Orlicz space with Luxemburg norm and $D_t$ is the operator of differentition at the point $t$. As an application, we state necessary and sufficient conditions in the Kolmogorov problem for three numbers. Also we solve the Stechkin problem, i.e., the problem of approximating an unbounded operator of differentition $D_t$ by bounded linear operators for the class of functions $x$ such that $\|x''\|_{L_{N}^*[0,1]}\le 1$.
Keywords: Kolmogorov problem for three numbers, Stechkin problem, Orlicz space, operator of differentition, Banach space, modulus of continuity.
Mots-clés : Luxemburg norm 
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     title = {The {Kolmogorov} and {Stechkin} {Problems} for {Classes} of {Functions} {Whose} {Second} {Derivative} {Belongs} to the {Orlicz} {Space}},
     journal = {Matemati\v{c}eskie zametki},
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Yu. V. Babenko; D. Skorokhodov. The Kolmogorov and Stechkin Problems for Classes of Functions Whose Second Derivative Belongs to the Orlicz Space. Matematičeskie zametki, Tome 91 (2012) no. 2, pp. 172-183. http://geodesic.mathdoc.fr/item/MZM_2012_91_2_a1/