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Erschienen in:
2010 | OriginalPaper | Buchkapitel
Factorization of matrix functions analytic in a strip
verfasst von : Harm Bart, Marinus A. Kaashoek, André C. M. Ran
Erschienen in: A State Space Approach to Canonical Factorization with Applications
Verlag: Birkhäuser Basel
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This chapter deals with
m × m
matrix-valued functions of the form
5.1
$$ W(\lambda ) = I - \int_{ - \infty }^\infty {e^{i\lambda t} k(t)dt,} $$
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
5.2
$$ k(t) = e^{\omega |t|} h(t) with h \in L_1^{m \times m} (\mathbb{R}). $$
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).