The chain rule for F-differentiation

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

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.

Citation

Chaobankoh, T., Feinstein, J., & Morley, S. (2016). The chain rule for F-differentiation. Bulletin of the Irish Mathematical Society,

• 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
• Contract Date: Jul 19, 2016