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Proceedings of the American Mathematical Society
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
MathSciNet review: 619986
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0619986-3
PII: S 0002-9939(1981)0619986-3
Keywords: Discrete differences, $ k$-discrete Gâteaux and Fréchet differentials, $ n$-convex functions
Article copyright: © Copyright 1981 American Mathematical Society