## Arkiv för Matematik

• Ark. Mat.
• Volume 7, Number 5 (1968), 433-441.

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

Yngve Domar

#### Abstract

LetA 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

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.

#### Article information

Source
Ark. Mat., Volume 7, Number 5 (1968), 433-441.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485893675

Digital Object Identifier
doi:10.1007/BF02590991

Mathematical Reviews number (MathSciNet)
MR234216

Zentralblatt MATH identifier
0165.48801

Rights

#### Citation

Domar, Yngve. An extremal problem related to Kolmogoroff’s inequality for bounded functions. Ark. Mat. 7 (1968), no. 5, 433--441. doi:10.1007/BF02590991. https://projecteuclid.org/euclid.afm/1485893675

#### References

• Achiezer, N. I., Vorlesungen über Approximationstheorie. Berlin 1953 (1947).
• Bang, T., Une inégalité de Kolmogoroff et les functions presqueperiodiques. Danske Vid. Selsk Mat.-Fys. Medd. XIX, 4 (1941).
• Domar, Y., On the uniqueness of minimal extrapolations. Ark. Mat. 4, 19–29 (1959).
• Herz, C. S., The spectral theory of bounded functions. Trans. Am. Math. Soc. 94, 181–232 (1960).
• Hörmander, L., A new proof and a generalization of an inequality of Bohr. Math. Scand. 2, 33–45 (1954).
• Kolmogoroff, A. N., On inequalities between upper bounds of consecutive derivatives of an arbitrary function defined on an infinite interval. Učenye Zapiski Moskov. Gos. Univ. Matematika 30, 3–16 (1939); Amer. Math. Soc. Translation 4.