An analog of the Hille theorem for hypercomplex functions in a finite-dimensional commutative algebra

TL;DR

The paper proves an analog of the Hille theorem for hypercomplex functions valued in a finite-dimensional commutative Banach algebra \mathbb{A}_n^m: a function that is locally bounded and differentiable in the sense of Gâteaux on a domain \Omega \subset E_k is also differentiable in the sense of Lorch, and equivalently monogenic. This is achieved under the Ek condition and relies on a Cartan-basis framework, a CR-type analysis of complex projections, and an integral representation in terms of holomorphic data on these projections. A constructive description expresses such functions via contour integrals with holomorphic inputs, yielding a finite-dimensional, hypercomplex-analytic decomposition. The results extend prior work for lower-dimensional algebras (k=3) to general finite-dimensional commutative Banach algebras, linking local boundedness, monogenicity, and Lorch differentiability in this setting.

Abstract

We prove that a locally bounded and differentiable in the sense of Gateaux function given in a finite-dimensional commutative Banach algebra over the complex field is also differentiable in the sense of Lorch.