Factorization of matrix functions analytic in a strip

Abstract

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).