Real Analysis Exchange

Higher Order Uniform Smoothness and Differentiability of Real Functions

Matteo Rocca and Davide La Torre

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Abstract

It is known that smoothness-type conditions have several implications on continuity and differentiability properties of real functions. When these conditions hold uniformly on an interval the implications become even stronger. The aim of this paper is to extend to higher orders the relations between uniform smoothness-type conditions and differentiability, taking into account higher order divided differences.

Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 657-668.

Dates
First available in Project Euclid: 27 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1214571358

Mathematical Reviews number (MathSciNet)
MR1844144

Zentralblatt MATH identifier
1015.26014

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A16: Lipschitz (Hölder) classes

Keywords
H\"older functions divided differences

Citation

Torre, Davide La; Rocca, Matteo. Higher Order Uniform Smoothness and Differentiability of Real Functions. Real Anal. Exchange 26 (2000), no. 2, 657--668. https://projecteuclid.org/euclid.rae/1214571358


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References

  • H. Brezis, Analyse fonctionelle- Theorie et applications. Masson Editeur, Paris, 1963.
  • P. S. Bullen, A criterion for n-convexity. Pac. J. Math., 36 (1971), 81–98.
  • Z. Buczolich, M. J. Evans, P. D. Humke, Approximate high order smoothness. Acta Math. Hung., 61 (1993), 369–388.
  • T. K. Dutta, Generalized smooth functions. Acta Math. Acad. Sci. Hungar., 40 (1982), 29–37.
  • T. K. Dutta, N. Mukhopadhyay, Generalized smooth functions II. Acta. Math. Hung., 55 (1990), 47–56.
  • M. J. Evans, C. E. Weil, Peano derivatives: a survey. Real Analysis Exchange, 7 (1981-82), 5–23.
  • M. J. Evans, High order smoothness. Acta Math. Hung., 50 (1987), 17–20.
  • I. Ginchev, A. Guerraggio, M. Rocca, Equivalence of Peano and Riemann derivatives. Proceedings of the Workshop on Optimization and Generalized Convexity for economic applications, Verona, may 1998, (G. Giorgi and F.A. Rossi eds.)
  • I. Ginchev, M. Rocca, On Peano and Riemann derivatives. Rendiconti del Circolo Matematico di Palermo, 49, (2000), 153–164.
  • I. Ginchev, A. Guerraggio, M. Rocca, Equivalence of $(n+1)$-th order Peano and usual derivatives for n-convex functions. Real Analysis Exchange, 25 (2000), 513–520.
  • A. Guerraggio, M. Rocca, Derivate dirette di Riemann e di Peano. Convessitá e Calcolo Parallelo (G. Giorgi and F.A. Rossi eds.), Verona, 1997.
  • E. Isaacson, B. H. Keller, Analysis of numerical methods. Wiley, New York, 1966.
  • D. La Torre, M. Rocca, $C^{k,1}$ functions and Riemann derivatives. Real Analysis Exchange, 25, (2000), 743–752.
  • J. Marcinkiewicz, A. Zygmund, On the differentiability of functions and summability of trigonometrical series. Fund. Math., 26 (1936), 1–43.
  • H. W. Oliver, The exact Peano derivative. Trans. Amer. Math.Soc., 76 (1954), 444–456.
  • G. Peano, Sulla formula di taylor. Atti Accad. Sci. Torino, 27 (1981-92), 40–46.
  • B. S. Thomson, Symmetric properties of real functions. Marcel Dekker, New York, 1994.
  • C. E. Weil, The Peano notion of higher order differentiation. Math. Japon., 42 (1995), 587–600.
  • A. Zygmund, Trigonometric Series. Cambridge University Press, Cambridge, 1959.