Real Analysis Exchange

C{k,1} Functions and Riemann Derivatives

Davide La Torre and Matteo Rocca

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Abstract

In this work we provide a characterization of $C^{k,1}$ functions of one real variable (that is, $k$ times differentiable with locally Lipschitz $k$-th derivative) by means of $(k+1)$-th divided differences and Riemann derivatives. In particular we prove that the class of $C^{k,1}$ functions is equivalent to the class of functions with bounded $(k+1)$-th divided difference. From this result we deduce a Taylor's formula for this class of functions and a characterization through Riemann derivatives.

Article information

Source
Real Anal. Exchange, Volume 25, Number 2 (1999), 743-752.

Dates
First available in Project Euclid: 3 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.rae/1230995409

Mathematical Reviews number (MathSciNet)
MR1778527

Zentralblatt MATH identifier
1016.26007

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

Keywords
Riemann derivatives divided differences Lipschitz functions

Citation

La Torre, Davide; Rocca, Matteo. C { k ,1} Functions and Riemann Derivatives. Real Anal. Exchange 25 (1999), no. 2, 743--752. https://projecteuclid.org/euclid.rae/1230995409


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References

  • H. Brezis, Analyse fonctionelle- Theorie et applications. Masson Editeur, Paris, 1963.
  • M. J. Evans, C. E. Weil, Peano derivatives: a survey. Real Analysis Exchange, 7, 1981-82, 5–23.
  • 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, F. A. Rossi eds.)
  • I. Ginchev, M. Rocca M, On Peano and Riemann derivatives, Rendiconti del Circolo Matematico di Palermo, to appear.
  • I. Ginchev, A. Guerraggio, M. Rocca, Equivalence of $(n+1)$-th order Peano and usual derivatives for $n$-convex functions., Real Analysis Exchange, this issue.
  • A. Guerraggio, M, Rocca, Derivate dirette di Riemann e di Peano. Convessitá e Calcolo Parallelo, Verona 1997.
  • J. B. Hiriart-Urruty, J. J. Strodiot, V. Hien Nguyen, Generalized Hessian matrix and second order optimality conditions for problems with $C^{1,1}$ data. Appl. Math. Optim., 11, 1984, 43–56.
  • E. Isaacson, B. H. Keller, Analysis of numerical methods. Wiley, New York, 1966.
  • D. La Torre, M. Rocca, Weak derivatives, optimization and convex function theory. Internal Report, 47, Institute of Quantitative Methods, "Bocconi " University of Milan, 1999.
  • D. T. Luc, Taylor's formula for $C^{k,1}$ functions. SIAM J. Optimization, 5, 1995, 659–669.
  • J. Marcinkiewicz, A. Zygmund, On the differentiability of functions and summability of trigonometric 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, 1891-92, 40–46.