# An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

Fundamenta Mathematicae (1996)

• Volume: 151, Issue: 1, page 21-38
• ISSN: 0016-2736

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## Abstract

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Let f be a measurable function such that ${\Delta }_{k}\left(x,h;f\right)={O\left(|h|}^{\lambda }\right)$ at each point x of a set E, where k is a positive integer, λ > 0 and ${\Delta }_{k}\left(x,h;f\right)$ 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 ${f}_{\left(k\right)}$ exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative ${f}_{\left(\left[\lambda \right]\right)}$ 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 ${f}_{\left(\lambda \right),a}$ exists on E then ${f}_{\left(\lambda \right)}$ exists a.e. on E.

## How to cite

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Mukhopadhyay, S., and Mitra, S.. "An extension of a theorem of Marcinkiewicz and Zygmund on differentiability." Fundamenta Mathematicae 151.1 (1996): 21-38. <http://eudml.org/doc/212180>.

abstract = {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.},
author = {Mukhopadhyay, S., Mitra, S.},
journal = {Fundamenta Mathematicae},
keywords = {measurable function; Peano derivative; approximate Peano derivative},
language = {eng},
number = {1},
pages = {21-38},
title = {An extension of a theorem of Marcinkiewicz and Zygmund on differentiability},
url = {http://eudml.org/doc/212180},
volume = {151},
year = {1996},
}

TY - JOUR
AU - Mitra, S.
TI - An extension of a theorem of Marcinkiewicz and Zygmund on differentiability
JO - Fundamenta Mathematicae
PY - 1996
VL - 151
IS - 1
SP - 21
EP - 38
AB - 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.
LA - eng
KW - measurable function; Peano derivative; approximate Peano derivative
UR - http://eudml.org/doc/212180
ER -

## References

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1. [1] P. S. Bullen and S. N. Mukhopadhyay, Peano derivatives and general integrals, Pacific J. Math. 47 (1973), 43-58. Zbl0252.26002
2. [2] A. Denjoy, Sur l'intégration des coefficients différentielles d'ordre supérieur, Fund. Math. 25 (1935), 273-326. Zbl61.1115.03
3. [3] H. Fejzic and C. E. Weil, Repairing the proof of a classical differentiation result, Real Anal. Exchange 19 (1993-94), 639-643. Zbl0818.26002
4. [4] J. Marcinkiewicz, Sur les séries de Fourier, Fund. Math. 27 (1937), 38-69.
5. [5] J. Marcinkiewicz and A. Zygmund, On the differentiability of functions and summability of trigonometric series, Fund. Math. 26 (1936), 1-43. Zbl0014.11102
6. [6] S. N. Mukhopadhyay and S. Mitra, Measurability of Peano derivates and approximate Peano derivates, Real Anal. Exchange 20 (1994-95), 768-775. Zbl0832.26002
7. [7] S. Saks, Theory of the Integral, Dover, 1964.
8. [8] E. M. Stein and A. Zygmund, On the differentiability of functions, Studia Math. 23 (1964), 247-283. Zbl0122.30203
9. [9] A. Zygmund, Trigonometric Series I, II, Cambridge Univ. Press, 1968. Zbl0628.42001

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