### abstract

- We give a simple constructive proof of a factorization theorem for scalar rational functions with a non-negative real part on the imaginary axis. A Mathematica program, performing this factorization, is provided … The main result of this paper is a constructive proof of a factorization theorem for rational functions with a non-negative real part on the imaginary axis; see Theorem 1.1 below. To state the result we first need some definitions … (1.4) Re p ( s ) ≥ 0 , s ∈ i R . … One can weaken condition (1.1) and assume that a function p is only of bounded type in C + and satisfies (1.4) almost everywhere on the imaginary line. These functions will be called generalized positive and are thus denoted by GP … An intermediate case is when p is meromorphic in C + and the associated kernel K p has a finite number of negative squares, denoted by κ . Recall that this means that for every choice of positive integer n , and every choice of …