An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

S. Mukhopadhyay¹ ; S. Mitra¹

¹

Fundamenta Mathematicae, Tome 151 (1996) no. 1, pp. 21-38.

Résumé

Let f be a measurable function such that $Δ_k(x,h;f) = O(|h|^λ)$ at each point x of a set E, where k is a positive integer, λ > 0 and $Δ_k(x,h;f)$ is the symmetric difference of f at x of order k. Marcinkiewicz and Zygmund [5] proved that if λ = k and if E is measurable then the Peano derivative $f_{(k)}$ exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative $f_{([λ])}$ exists a.e. on E and that the result is false if λ = k-1; it is further proved that if λ is any positive integer and if the approximate Peano derivative $f_{(λ),a}$ exists on E then $f_{(λ)}$ exists a.e. on E.

Détail

DOI : 10.4064/fm-151-1-21-38