An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

Mukhopadhyay, S.; Mitra, S.

Fundamenta Mathematicae, Tome 149 (1996), p. 21-38 / Harvested from The Polish Digital Mathematics Library

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.

Publié le : 1996-01-01

Zbl 0869.26002

EUDML-ID : urn:eudml:doc:212180

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     author = {S. Mukhopadhyay and S. Mitra},
     title = {An extension of a theorem of Marcinkiewicz and Zygmund on differentiability},
     journal = {Fundamenta Mathematicae},
     volume = {149},
     year = {1996},
     pages = {21-38},
     zbl = {0869.26002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv151i1p21bwm}
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Mukhopadhyay, S.; Mitra, S. An extension of a theorem of Marcinkiewicz and Zygmund on differentiability. Fundamenta Mathematicae, Tome 149 (1996) pp. 21-38. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv151i1p21bwm/

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