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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by S. Dayal

Proc. Amer. Math. Soc. 83 (1981), 77-82

DOI: https://doi.org/10.1090/S0002-9939-1981-0619986-3

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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.

References

J. Marshall Ash, A characterization of the Peano derivative, Trans. Amer. Math. Soc. 149 (1970), 489–501. MR 259041, DOI 10.1090/S0002-9947-1970-0259041-5
P. L. Butzer and W. Kozakiewicz, On the Riemann derivatives for integrable functions, Canad. J. Math. 6 (1954), 572–581. MR 64131, DOI 10.4153/cjm-1954-062-5
Z. Ciesielski, Some properties of convex functions of higher orders, Ann. Polon. Math. 7 (1959), 1–7. MR 109202, DOI 10.4064/ap-7-1-1-7

On local representations of functions on normed linear spaces

S. Dayal, A converse of Taylor’s theorem for functions on Banach spaces, Proc. Amer. Math. Soc. 65 (1977), no. 2, 265–273. MR 448394, DOI 10.1090/S0002-9939-1977-0448394-2

Sur l’intégration des coefficients différentiels d’ordre supérieur

25

J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319

Linear operators

Roman Ger, Convex functions of higher orders in Euclidean spaces, Ann. Polon. Math. 25 (1971/72), 293–302. MR 304580, DOI 10.4064/ap-25-3-293-302
Roman Ger, $n$-convex functions in linear spaces, Aequationes Math. 11 (1974), 172–176. MR 358119, DOI 10.1007/BF01832854
M. Z. Nashed, On the representation and differentiability of operators, Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969) Akadémiai Kiadó, Budapest, 1972, pp. 325–330. MR 0390766

Differentiability and relative properties of nonlinear operators: Some aspects of the role of differentials in nonlinear analysis

Clifford E. Weil, On approximate and Peano derivatives, Proc. Amer. Math. Soc. 20 (1969), 487–490. MR 233944, DOI 10.1090/S0002-9939-1969-0233944-7