**Cristina Camara**

Universidade Técnica de Lisboa

The Fredholmness and invertibility of
finite-interval convolution operators acting on spaces ,
where is a compact interval in , is closely
related to the Wiener-Hopf factorization of its matrix-valued
symbol. This factorization is studied for some classes of symbols
whose entries are almost-periodic polynomials with Fourier
spectrum in the group
). The
factorization problem is solved by calculating one solution to the
Riemann-Hilbert problem
in
and obtaining a second linearly
independent solution by means of an appropriate transformation on
the space of solutions of the Riemann-Hilbert problem. Some
unexpected, but interesting, results are obtained regarding the
Fourier spectrum of the solutions of this class of Riemann-Hilbert
problems. The Wiener-Hopf factors of are explicitly obtained,
which allow us to establish invertibility criteria and formulas
for the inverse of the associated operator.

2005-05-23