Factorization of a class of almost-periodic triangular symbols and related Riemann-Hilbert problems

Cristina Camara

Universidade Técnica de Lisboa

The Fredholmness and invertibility of finite-interval convolution operators acting on spaces $L_2(I)$, where $I$ is a compact interval in $\mathbb{R}$, is closely related to the Wiener-Hopf factorization of its matrix-valued symbol. This factorization is studied for some classes of symbols $G$ whose entries are almost-periodic polynomials with Fourier spectrum in the group $\alpha\mathbb{Z}+\beta\mathbb{Z}+\mathbb{Z}$ $\alpha,\beta\in ]0,1[,\;\frac{\alpha}{\beta}\not\in \mathbb{Q}$). The factorization problem is solved by calculating one solution to the Riemann-Hilbert problem $G\Phi_{+} = \Phi_{-}$ in $L_{\infty}(\mathbb{R})$ 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 $G$ are explicitly obtained, which allow us to establish invertibility criteria and formulas for the inverse of the associated operator.