Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$ k$-discrete differentials of certain operators on Banach spaces


Author: S. Dayal
Journal: Proc. Amer. Math. Soc. 83 (1981), 77-82
MSC: Primary 58C20; Secondary 26E99, 41A65, 49A51
DOI: https://doi.org/10.1090/S0002-9939-1981-0619986-3
MathSciNet review: 619986
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By observing a convex property of discrete differences, one-sided $ k$-discrete, $ k$-discrete Gâteaux and $ k$-discrete Fréchet differentials are introduced. It is proved that a locally bounded $ n$-convex function has $ k$-discrete Fréchet differentials for $ 1 \leqslant k \leqslant n - 2$ and one-sided $ (n - 1)$-discrete differentials at every point of its domain. Various properties of discrete differentials of an $ n$-convex function are studied. As an application of these results the author proves that an $ n$-convex function has a strong $ (n - 2)$-Taylor series expansion and an $ (n - 1)$th Fréchet differential provided it has a strong $ n$-Taylor series expansion about the point.


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

  • [1] J. M. Ash, A characterisation of the Peano derivatives, Trans. Amer. Math. Soc. 149 (1970), 489-501. MR 0259041 (41:3683)
  • [2] P. L. Butzer and W. Kozakiewez, On Riemann derivatives of integrable functions, Canad. J. Math. 6 (1954), 572-581. MR 0064131 (16:230f)
  • [3] Z. Ciesielski, Some properties of convex functions of higher order, Ann. Polon. Math. 7 (1959), 572-581. MR 0109202 (22:89)
  • [4] S. Dayal, On local representations of functions on normed linear spaces, Ph.D. thesis, Case Western Reserve University, Cleveland, Ohio, 1972.
  • [5] -, A converse of Taylor's theorem for functions on Banach spaces, Proc. Amer. Math. Soc. 65 (1977), 265-273. MR 0448394 (56:6701)
  • [6] A. Denjoy, Sur l'intégration des coefficients différentiels d'ordre supérieur, Fund. Math. 25 (1935), 273-362.
  • [7] J. Dieudonné, Foundations of modern analysis, Academic Press, New York, 1960. MR 0120319 (22:11074)
  • [8] N. Dunford and J. T. Schwartz, Linear operators. I, Interscience, New York, 1958.
  • [9] R. Ger, Convex functions of higher orders in Euclidean spaces, Ann. Polon. Math. 25 (1972), 293-302. MR 0304580 (46:3715)
  • [10] -, $ n$-convex functions in linear spaces, Aequationes Math. 10 (1974), 172-176. MR 0358119 (50:10584)
  • [11] M. Z. Nashed, On the representation and differentiability of operators, Proceedings Conf. Constructive Theory of Functions (Budapest, 1969), Akad. Kiadó, Budapest, 1972, pp. 325-330. MR 0390766 (52:11589)
  • [12] -, Differentiability and relative properties of nonlinear operators: Some aspects of the role of differentials in nonlinear analysis, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, 1970), Academic Press, New York, 1971, pp. 103-309.
  • [13] C. E. Weil, On approximation and Peano derivatives, Proc. Amer. Math. Soc. 20 (1969), 487-490. MR 0233944 (38:2265)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58C20, 26E99, 41A65, 49A51

Retrieve articles in all journals with MSC: 58C20, 26E99, 41A65, 49A51


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0619986-3
Keywords: Discrete differences, $ k$-discrete Gâteaux and Fréchet differentials, $ n$-convex functions
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society