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The chain rule for F-differentiation

Chaobankoh, T.; Feinstein, Joel; Morley, S.

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Authors

T. Chaobankoh

S. Morley



Abstract

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.

Journal Article Type Article
Acceptance Date May 9, 2016
Online Publication Date Jun 1, 2016
Publication Date Jun 1, 2016
Deposit Date Jul 19, 2016
Publicly Available Date Jul 19, 2016
Journal Irish Mathematical Society Bulletin
Electronic ISSN 0791-5578
Peer Reviewed Peer Reviewed
Issue 77
Public URL https://nottingham-repository.worktribe.com/output/979839
Publisher URL http://www.maths.tcd.ie/pub/ims/bull77/Feinstein.pdf

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