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Mean value theorems for generalized Riemann derivatives


Authors: J. M. Ash and R. L. Jones
Journal: Proc. Amer. Math. Soc. 101 (1987), 263-271
MSC: Primary 26A24; Secondary 65D25
DOI: https://doi.org/10.1090/S0002-9939-1987-0902539-2
MathSciNet review: 902539
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Abstract: Let $ x,e \geqslant 0,{u_0} < \cdots < {u_{d + e}}$ and $ h > 0$ be real numbers. Let $ f$ be a real valued function and let $ \Delta (h;u,w)f(x){h^{ - d}}$ be a difference quotient associated with a generalized Riemann derivative. Set $ I = (x + {u_0}h,x + {u_{d + e}}h)$ and let $ f$ have its ordinary $ (d - 1)$st derivative continuous on the closure of $ I$ and its $ d$th ordinary derivative $ {f^{(d)}}$ existent on $ I$. A necessary and sufficient condition that a difference quotient satisfy a mean value theorem (i.e., that there be a $ \xi \in I$ such that the difference quotient is equal to $ {f^{(d)}}(\xi ))$ is given for $ d = 1$ and $ d = 2$. The condition is sufficient for all $ d$. It is used to show that many generalized Riemann derivatives that are "good" for numerical analysis do not satisfy this mean value theorem.


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 0204583 (34:4422)
  • [2] -, A characterization of the Peano derivative, Trans. Amer. Math. Soc. 149 (1970), 489-501. MR 0259041 (41:3683)
  • [3] -, Very generalized Riemann derivatives, generalized Riemann derivatives and associated summability methods, Real Anal. Exchange 11 (1985-86), 10-29. MR 828476
  • [4] J. M. Ash and R. L. Jones, Optimal numerical differentiation using three function evaluations, Math. Comp. 37 (1981), 159-167. MR 616368 (84a:65008)
  • [5] J. M. Ash, S. Janson, and R. L. Jones, Optimal numerical differentiation using n function evaluations, Estrato da Calcolo 21 (1984), 151-169. MR 799618 (86k:65017)
  • [6] G. Cantor, Gesammelte Abhandlungen, Georg Olms, Hildesheim, 1962, 80-83. Beweis, das eine für jeden reellen Wert von x durch eine trigonometrische Reihe gegebene Funktion $ f(x)$ sich nur auf eine einzige Weise in dieser Form darstellen lässt, Crelles J. Math. 72 (1870), 139-142. MR 0148517 (26:6024)
  • [7] A. Denjoy, Sur l'intégration des coefficients différentiels d'ordre supérieur, Fund. Math. 25 (1935), 273-326.
  • [8] E. Isaacson and H. B. Keller, Analysis of numerical methods, Wiley, New York; 1966. MR 0201039 (34:924)
  • [9] P. J. O'Connor, Generalized differentiation of functions of a real variable, Ph.D. dissertation, Wesleyan Univ., Middletown, Conn., 1969.
  • [10] B. Riemann, Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe, Ges. Werke, 2. Aufl., Leipzig, 1892, pp. 227-271; also Dover, New York, 1953.
  • [11] A. Zygmund, Trigonometric series, Vols. I and II, Cambridge Univ. Press, Cambridge, 1959. MR 0236587 (38:4882)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0902539-2
Keywords: Generalized Riemann derivative, mean value theorem
Article copyright: © Copyright 1987 American Mathematical Society

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