The Regularization of Unitary Matrix Functions

Background. The limit behavior at infinity of unitary matrix functions of real argument and arbitrary finite dimension is considered. The question of regular variation in Karamata sense of these functions is studied. Objective. The main aim of this work is to find the conditions under which not regularly varying unitary matrix function of real argument can be regularized by variable substitution. Methods. It is shown in the paper that substitution t→logt t→log?t converts two-dimensional unitary matrix function into a power function with known matrix degree. This property is the base of all main result's proofs. Results. The conditions under which the unitary matrix function of arbitrary finite dimension can be regularized are obtained. Conclusions. A significant difference between the matrix functions of dimension lower than 4 and the matrix functions of higher dimension is established. The constructed example of unitary matrix function of 4-dimension that can't be regularized by substitution t→logt t→log?t shows this difference.