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Transactions of the American Mathematical Society

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A characterization of the Peano derivative

Author: J. Marshall Ash
Journal: Trans. Amer. Math. Soc. 149 (1970), 489-501
MSC: Primary 26.43
MathSciNet review: 0259041
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Abstract: For each choice of parameters $ \{ {a_i},{b_i}\} ,i = 0,1, \ldots ,n + e$, satisfying certain simple conditions, the expression

$\displaystyle \mathop {\lim }\limits_{h \to 0} {h^{ - n}}\sum\limits_{i = 0}^{n + e} {{a_i}f(x + {b_i}h)} $

yields a generalized nth derivative. A function f has an nth Peano derivative at x if and only if all the members of a certain subfamily of these nth derivatives exist at x. The result holds for the corresponding $ {L^p}$ derivatives. A uniformity lemma in the proof (Lemma 2) may be of independent interest.

Also, a new generalized second derivative is introduced which differentiates more functions than the ordinary second derivative but fewer than the second Peano derivative.

References [Enhancements On Off] (What's this?)

  • [1] J. M. Ash, Generalizations of the Riemann derivative, Trans. Amer. Math. Soc. 126 (1967), 181-199. MR 34 #4422. MR 0204583 (34:4422)
  • [2] A. Denjoy, Sur l'intégration des coefficients différentiels d'ordre supérieur, Fund. Math. 25 (1935), 273-326.
  • [3] -, Leçons sur le calcul des coefficients d'une séries trigonometrique, Tome I: La différentiation seconde mixte et son application aux séries trigonometriques, Gauthier-Villars, Paris, 1941, p. 15. MR 8, 260.
  • [4] J. Korevaar, T. van Aardenne-Ehrenfest and N. G. deBruijn, A note on slowly oscillating functions, Nieuw Arch. Wisk. (2) 23 (1949), 77-86. MR 10, 358. MR 0027812 (10:358b)
  • [5] J. Marcinkiewicz and A. Zygmund, On the differentiability of functions and summability of trigonometric series, Fund. Math. 26 (1936), 1-43.
  • [6] P. T. O'Connor, Generalized differentiation of functions of a real variable, Doctoral Dissertation, Wesleyan University, Middletown, Ct., 1969.
  • [7] E. C. Titchmarsh, The theory of functions, 2nd ed., Oxford Univ. Press, London, 1939, p. 355. MR 0197687 (33:5850)

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Keywords: Peano derivative, generalized derivative, Riemann derivative, function of one real variable, $ {L^p}$ derivative
Article copyright: © Copyright 1970 American Mathematical Society

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