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

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.