On measurable functions with vanishing differences

Marek Kuczma

Published: 1992-09-30

Vol. 6 (1992)

On measurable functions with vanishing differences

Marek Kuczma

Abstract

It is shown (under suitable conditions on H⊂R) that if f: R→R is a measurable function such that for an n∈N₀ and every h∈H we have Δhⁿ⁺¹f(x) = 0 almost everywhere on R, then f is equal almost everywhere on R to a polynomial of degree at most n. In particular, every measurable polynomial function f: R→R is a polynomial. In fact, these (essentially known) results are here proved in a more general and more abstract form. The paper contains also a version of the Łomnicki-type theorem on measurable microperiodic functions.

Download files

PDF

Citation rules

Kuczma, M. (1992). On measurable functions with vanishing differences. Annales Mathematicae Silesianae, 6, 42–60. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14266

References

1. J.A. Baker, Functional equations, tempered distributions and Fourier transforms, Trans. Amer. Math. Soc. 315 (1989), 57-68.
2. R.P. Boas Jr., Functions which are odd about several points, Nieuw Arch. Wisk. (3) 1 (1953), 27-32.
3. Z. Ciesielski, Some properties of convex functions of higher orders, Ann. Polon. Math. 7 (1959), 1-7.
4. R. Engelking, General topology, Monografie Mat. 60, Polish Scientific Publishers, Warszawa 1977.
5. R. Ger, On almost polynomial functions, Colloq. Math. 24 (1971), 95-101.
6. T. Husain, Introduction to topological group, W.B. Saunders, Philadelphia-London, 1966.
7. J.H.B. Kemperman, A general funcional equation, Trans. Amer. Math. Soc. 86 (1957), 28-56.
8. Z. Kominek, M. Kuczma, Theorem of Bernstein-Doetsch, Piccard and Mehdi, and semilinear topology, Arch. Math. (Basel) 51 (1988), 1-8.
9. M. Kuczma, An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality, Polish Scientific Publishers and Silesian University Press, Warszawa-Kraków-Katowice, 1985.
10. M. Kuczma, Note on microperiodic functions, Radovi Mat. 5 (1989), 127-140.
11. S. Mazur, W. Orlicz, Grundlegende Eigenschaften der polynomischen Operationen I-II, Studia Math. 5 (1934), 50-68 and 179-189.
12. W. Sierpiński, Hypothèse du continue, Warszawa, 1934. Google Scholar

Vol. 6 (1992)
Published: 1992-09-30

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

Submit