Factorization of matrix functions analytic in a strip

This chapter deals with m × m matrix-valued functions of the form $$ W(\lambda ) = I - \int_{ - \infty }^\infty {e^{i\lambda t} k(t)dt,} $$ (5.1) where k is an m × m matrix-valued function with the property that for some ω < 0 the entries of e−ω|t|k(t) are Lebesgue integrable on the real line. In other words, k is of the form $$ k(t) = e^{\omega |t|} h(t) with h \in L_1^{m \times m} (\mathbb{R}). $$ (5.2) It follows that the function W is analytic in the strip \( \left| {\mathfrak{F}\lambda } \right| \), where τ=−ω. This strip contains the real line. The aim is to extend the canonical factorization theorem of Chapter 5 to functions of the type (5.1).