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

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2015

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Mathematical Analysis and Applications

ISSN

0022-247X

e-ISSN

—

Svazek periodika

423

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

14

Strana od-do

594-607

Kód UT WoS článku

000349706000035

EID výsledku v databázi Scopus

2-s2.0-84923046355