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A converse of Taylor's theorem for functions on Banach spaces

Author: S. Dayal
Journal: Proc. Amer. Math. Soc. 65 (1977), 265-273
MSC: Primary 58C20; Secondary 46G05
MathSciNet review: 0448394
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Abstract: The main results are the local representation theorems associating the local weak n-Taylor series expansion of a function defined on a Banach space to a local n-Taylor series expansion of the coefficients. These theorems are used to prove a converse of Taylor's theorem which uses weaker hyptohesis than used by others. Another useful application of the above results is done in [2] to study a class of functions called n-convex functions.

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

  • [1] R. Abraham and J. Robin, Transversal mappings and flows, Benjamin, New York, 1967. MR 39 #2181. MR 0240836 (39:2181)
  • [2] S. Dayal, Local representation and higher differentiability of n-convex operators on Banach spaces (to appear).
  • [3] -, Local representation of functions on normed linear spaces, Ph.D Thesis, Case Western Reserve Univ., Cleveland, Ohio, 1972.
  • [4] -, A converse of Taylor's theorem for functions on locally convex and topological linear spaces (to appear).
  • [5] J. Dieudonné, Foundations of modern analysis, Academic Press, New York, 1960. MR 22 #11074. MR 0120319 (22:11074)
  • [6] G. Glasser, Etude de quelques algèbres Tayloriennes, J. Analyse Math. 6 (1958), 1-124. MR 0101294 (21:107)
  • [7] Francis B. Hilderbrand, Introduction to numerical analysis, McGraw-Hill, New York, 1956. MR 17, 788. MR 0075670 (17:788d)
  • [8] J. Kopeć and J. Musielak, On the estimation of the norm of the n-linear symmetric operators, Studia Math. 15 (1955), 29-30. MR 17, 512. MR 0073944 (17:512a)
  • [9] E. B. Leach, A note on inverse function theorems, Proc. Amer. Math. Soc. 12 (1961), 694-697. MR 23 #A3442. MR 0126146 (23:A3442)
  • [10] J. Marcinkiewicz and A. Zygmund, On the differentiability of functions and summability of trignometric series, Fund. Math. 26 (1936), 1-43.
  • [11] R. M. McLeod, Mean value theorems for vector valued functions, Proc. Edinburgh Math. Soc. (2) 14 (1964/65), 197-229. MR 32 #2522. MR 0185052 (32:2522)
  • [12] M. Z. Nashed, Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970), Academic Press, New York, 1971, pp. 103-309. MR 43 #2580. MR 0276840 (43:2580)
  • [13] A. E. Taylor, Addition to the theory of polynomials in normed linear spaces, Tôhoku Math. J. 44 (1938), 302-318.

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Keywords: Fréchet differential, strong differential, multilinear functions, discrete differences, convex property, convex set, local representations, weak and strong n-Taylor series expansion
Article copyright: © Copyright 1977 American Mathematical Society

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