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Factorization in commutative Banach algebras (2021)
Journal Article
Dales, H. G., Feinstein, J. F., & Pham, H. L. (2021). Factorization in commutative Banach algebras. Studia Mathematica, 259, 79-120. https://doi.org/10.4064/sm191216-22-7

Let A be a (non-unital) commutative Banach algebra. We consider when A has a variety of factorization properties: we list the (ob-vious) implications between these properties, and then consider whether any of these implications can be reversed in var... Read More about Factorization in commutative Banach algebras.

A general method for constructing essential uniform algebras (2018)
Journal Article
Feinstein, J., & Izzo, A. J. (2019). A general method for constructing essential uniform algebras. Studia Mathematica, 246, 47-61. https://doi.org/10.4064/sm170907-23-2

A general method for constructing essential uniform algebras with prescribed properties is presented. Using the method, the following examples are constructed: an essential, natural, regular uniform algebra on the closed unit disc; an essential, natu... Read More about A general method for constructing essential uniform algebras.

Quasianalyticity in certain Banach function algebras (2017)
Journal Article
Feinstein, J., & Morley, S. (in press). Quasianalyticity in certain Banach function algebras. Studia Mathematica, 238(2), https://doi.org/10.4064/sm8614-12-2016

Let X be a perfect, compact subset of the complex plane. We consider algebras of those functions on X which satisfy a generalized notion of differentiability, which we call F-differentiability. In particular, we investigate a notion of quasianalytici... Read More about Quasianalyticity in certain Banach function algebras.

Regularity points and Jensen measures for R(X) (2016)
Journal Article
Feinstein, J., & Yang, H. (in press). Regularity points and Jensen measures for R(X). Studia Mathematica, 235(1), https://doi.org/10.4064/sm8351-7-2016

We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in (2000). We show that, even for the natural uniform algebras R(X) (for compact plane... Read More about Regularity points and Jensen measures for R(X).

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

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 funct... Read More about The chain rule for F-differentiation.

Abstract Swiss cheese space and classicalisation of Swiss cheeses (2016)
Journal Article
Feinstein, J., Morley, S., & Yang, H. (2016). Abstract Swiss cheese space and classicalisation of Swiss cheeses. Journal of Mathematical Analysis and Applications, 438(1), 119-141. https://doi.org/10.1016/j.jmaa.2016.02.004

Swiss cheese sets are compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Such sets have provided numerous counterexamples in the theory of uniform algebras. In this paper, we introduce a topological... Read More about Abstract Swiss cheese space and classicalisation of Swiss cheeses.

Removability of exceptional sets for differentiable and Lipschitz functions (2015)
Journal Article
Craig, J., Feinstein, J., & Patrick, P. (2015). Removability of exceptional sets for differentiable and Lipschitz functions. 00 Journal not listed, 645, https://doi.org/10.1090/conm/645

We discuss removability problems concerning differentiability and pointwise Lipschitz conditions for functions of a real variable. We prove that, in each of the settings under consideration, a set is removable if and only if it has no perfect subsets... Read More about Removability of exceptional sets for differentiable and Lipschitz functions.

Swiss Cheeses and Their Applications (2015)
Presentation / Conference Contribution
Feinstein, J., Morley, S., & Yang, H. (2014, May). Swiss Cheeses and Their Applications. Presented at Seventh Conference on Function Spaces, Southern Illinois University at Edwardsville

Swiss cheese sets have been used in the literature as useful examples in the study of rational approximation and uniform algebras. In this paper, we give a survey of Swiss cheese constructions and related results. We describe some notable examples of... Read More about Swiss Cheeses and Their Applications.