𝑘-discrete differentials of certain operators on Banach spaces

Dayal, S.

doi:10.1090/S0002-9939-1981-0619986-3

$k$-discrete differentials of certain operators on Banach spaces
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Proc. Amer. Math. Soc. 83 (1981), 77-82 Request permission

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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