On measurable functions with vanishing differences



Abstract

It is shown (under suitable conditions on HR) that if f: RR is a measurable function such that for an nN0 and every hH 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: RR 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.


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Published : 1992-09-30


KuczmaM. (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

Marek Kuczma 
Instytut Matematyki, Uniwersytet Śląski w Katowicach  Poland



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