An extension of a theorem of Marcinkiewicz and Zygmund on differentiability
Fundamenta Mathematicae, Tome 151 (1996) no. 1, pp. 21-38
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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.
@article{10_4064_fm_151_1_21_38,
author = {S. Mukhopadhyay and S. Mitra},
title = {An extension of a theorem of {Marcinkiewicz} and {Zygmund} on differentiability},
journal = {Fundamenta Mathematicae},
pages = {21--38},
publisher = {mathdoc},
volume = {151},
number = {1},
year = {1996},
doi = {10.4064/fm-151-1-21-38},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm-151-1-21-38/}
}
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S. Mukhopadhyay; S. Mitra. An extension of a theorem of Marcinkiewicz and Zygmund on differentiability. Fundamenta Mathematicae, Tome 151 (1996) no. 1, pp. 21-38. doi: 10.4064/fm-151-1-21-38
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