Classes of functions in the set of circulant matrices and their characterization

In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of circulant matrices with complex entries. For these new classes of functions, we extend the recently obtained characterization theorems in [Li, B. Q. Two elementary properties of entire functions and their applications. Amer. Math. Monthly 122(2) (2015), 169--172] and [Li, B. Q. A characterization of rational functions. Amer. Math. Monthly. 132(3) (2025), 269--271] to an algebraic structure of circulant matrices that includes several complex variables. Our characterization theorems lead to the generalization of the variant of the fundamental theorem of algebra established recently in [Abramov, V. M. Extension of the fundamental theorem of algebra for polynomial matrix equation with circulant matrices. Amer. Math. Monthly. DOI: 10.1080/00029890.2024.2440298].