Convolution equations on finite intervals and factorization of matrix functions (Q1974669)

The authors consider the convolution equation \(\phi(t)+ \int^a_0 k_0(t- s)\phi(s) ds= f(t)\), where \(f(t)\in L_{n\times 1}(0,a)\), \(k_0\in L_{n\times n}(- a,a)\) are given and \(\phi\in L_{n\times 1}(0, a)\) is unknown. Here \(L_{n\times m}(a, b)\) is the space of \(n\times m\) matrix-valued functions with entries from \(L_1(a,b)\). This equation is associated with the matrix \[ A(\lambda)= \begin{pmatrix} -e^{-a\lambda\widehat k(\lambda)} & -I+\widehat k(\lambda)\\ I+\widehat k(\lambda) & -e^{- ia\lambda\widehat k(\lambda)}\end{pmatrix}, \] where \(\widehat k\) is the Fourier transform of any extension \(k\) to the real line of \(k_0\). Connections between the invertibility of the convolution operator \(B(\phi)(t)= \phi(t)+ \int^a_0 k_0(t- s)\phi(s) ds\) and the canonical factorization of the matrix \(A\) are given. Moreover, the inverse of \(B\) is explicitly expressed in terms of factorization of \(A\). The authors also deal with the discrete version.