The chain rule for F-differentiation

Chaobankoh, T. and Feinstein, Joel and Morley, S. (2016) The chain rule for F-differentiation. Irish Mathematical Society Bulletin (77). pp. 19-34. ISSN 0791-5578

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Let X be a perfect, compact subset of the complex plane, and let D (1)(X) denote the (complex) algebra of continuously complex-differentiable functions on X. Then D(1)(X) is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra D(1)(X), for certain sets X and collections F of paths in X, by considering F-differentiable functions on X.

In this paper, we investigate composition, the chain rule, and the quotient rule for this notion of differentiability. We give an example where the chain rule fails, and give a number of sufficient conditions for the chain rule to hold. Where the chain rule holds, we observe that the Fa a di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of F-differentiable functions.

Item Type: Article
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Depositing User: Eprints, Support
Date Deposited: 19 Jul 2016 08:57
Last Modified: 14 Sep 2016 06:11

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