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

Abstract

UDC 512.5

We introduce new classes of functions generalizing the well-known classes of functions of complex variable, such as the entire functions, meromorphic functions, rational functions, and polynomial functions, which take values in the set of circulant matrices with complex entries. For these new classes of functions, we extend some recently obtained  characterization theorems presented in [B. Q. Li,  Amer. Math. Monthly, 122, № 2, 169–172 (2015)] and [B. Q. Li, Amer. Math. Monthly, 132, № 3, 269–271 (2025)] to an algebraic structure of circulant matrices that includes several complex variables. Our characterization theorems  generalize a  recently established version of the fundamental theorem of algebra presented  in [V. M. Abramov, Amer. Math. Monthly, 132, № 4, 356–360 (2025)].