An extremal problem related to Kolmogoroff’s inequality for bounded functions

Yngve Domar

TBD mathscidoc:1701.332288

Arkiv for Matematik, 7, (5), 433-441, 1968.10
Let$A$and$B$be positive numbers and$m$and$n$positive integers,$m<n$. Then there is for complex valued functions φ on$R$with sufficient differentiability and boundedness properties a representationwhere$v$_{$1$}and$v$_{$2$}are bounded Borel measures with$v$_{$1$}absolutely continuous, such that there exists a function φ with ∣φ^{(n)}∣ ⩽$A$and ∣φ∣ ⩽$A$on$R$and satisfying $$\varphi ^{(m)} (0) = A\int_R {\left| {d\nu _1 } \right|} + B\int_R {\left| {d\nu _2 } \right|} .$$ This result is formulated and proved in a general setting also applicable to derivatives of fractional order. Necessary and sufficient conditions are given in order that the measures and the optimal functions have the same essential properties as those which occur in the particular case stated above.
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@inproceedings{yngve1968an,
  title={An extremal problem related to Kolmogoroff’s inequality for bounded functions},
  author={Yngve Domar},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203440662641097},
  booktitle={Arkiv for Matematik},
  volume={7},
  number={5},
  pages={433-441},
  year={1968},
}
Yngve Domar. An extremal problem related to Kolmogoroff’s inequality for bounded functions. 1968. Vol. 7. In Arkiv for Matematik. pp.433-441. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203440662641097.
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