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7 oktober 1968 An extremal problem related to Kolmogoroff’s inequality for bounded functions
Yngve Domar
Ark. Mat. 7(5): 433-441 (7 oktober 1968). DOI: 10.1007/BF02590991

Abstract

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 representation $$\varphi ^{(m)} = \varphi ^{(n)}{\large ×} \mspace{-17mu}{\large −} v_1 + \varphi {\large ×} \mspace{-17mu}{\large −} v_2,$$ where v1 and v2 are bounded Borel measures with v1 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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Yngve Domar. "An extremal problem related to Kolmogoroff’s inequality for bounded functions." Ark. Mat. 7 (5) 433 - 441, 7 oktober 1968. https://doi.org/10.1007/BF02590991

Information

Published: 7 oktober 1968
First available in Project Euclid: 31 January 2017

zbMATH: 0165.48801
MathSciNet: MR234216
Digital Object Identifier: 10.1007/BF02590991

Rights: 1968 © Almqvist & Wiksell

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Vol.7 • No. 5 • 7 oktober 1968
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