Smoothness via directional smoothness and Marchaud's theorem in Banach spaces

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F15%3A10317845" target="_blank" >RIV/00216208:11320/15:10317845 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1016/j.jmaa.2014.09.068" target="_blank" >http://dx.doi.org/10.1016/j.jmaa.2014.09.068</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.jmaa.2014.09.068" target="_blank" >10.1016/j.jmaa.2014.09.068</a>

Result language
angličtina
Original language name
Smoothness via directional smoothness and Marchaud's theorem in Banach spaces
Original language description
Classical Marchaud's theorem (1927) asserts that if f is a bounded function on [a, b], k is an element of N, and the (k + 1)th modulus of smoothness w(k+1) (f; t) is so small that eta(t) = integral(t)(0) omega(k+1)(f;s)/s(k+1) ds < +infinity for t > 0, then f is an element of C-k ((a, b)) and f((k)) is uniformly continuous with modulus C eta for some c > 0 (i.e. in our terminology f is C-k,C-c eta-smooth). Using a known version of the converse of Taylor theorem we easily deduce Marchaud's theorem for functions on certain open connected subsets of Banach spaces from the classical one-dimensional version. In the case of a bounded subset of R-n our result is more general than that of H. Johnen and K. Scherer (1973), which was proved by quite a different method. We also prove that if a locally bounded mapping between Banach spaces is C-k,C-w-smooth on every line, then it is C-k,C-w-smooth for some c > 0.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2015
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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