The chain rule for F-differentiation
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 investiga...
| Main Authors: | , , |
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| Format: | Article |
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Irish Mathematical Society
2016
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| Online Access: | https://eprints.nottingham.ac.uk/35158/ |
| _version_ | 1848795016491696128 |
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| author | Chaobankoh, T. Feinstein, Joel Morley, S. |
| author_facet | Chaobankoh, T. Feinstein, Joel Morley, S. |
| author_sort | Chaobankoh, T. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:25:23Z |
| format | Article |
| id | nottingham-35158 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:25:23Z |
| publishDate | 2016 |
| publisher | Irish Mathematical Society |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-351582020-05-04T20:05:06Z https://eprints.nottingham.ac.uk/35158/ The chain rule for F-differentiation Chaobankoh, T. Feinstein, Joel Morley, S. 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. Irish Mathematical Society 2016 Article PeerReviewed Chaobankoh, T., Feinstein, Joel and Morley, S. (2016) The chain rule for F-differentiation. Irish Mathematical Society Bulletin (77). pp. 19-34. ISSN 0791-5578 http://www.maths.tcd.ie/pub/ims/bull77/Feinstein.pdf |
| spellingShingle | Chaobankoh, T. Feinstein, Joel Morley, S. The chain rule for F-differentiation |
| title | The chain rule for F-differentiation |
| title_full | The chain rule for F-differentiation |
| title_fullStr | The chain rule for F-differentiation |
| title_full_unstemmed | The chain rule for F-differentiation |
| title_short | The chain rule for F-differentiation |
| title_sort | chain rule for f-differentiation |
| url | https://eprints.nottingham.ac.uk/35158/ https://eprints.nottingham.ac.uk/35158/ |