Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions

In this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated. We present necessary and sufficient conditions for the existence of extensions in terms of Toeplitz and Hankel operators on Besicovitch spaces and Lebesgue spaces. Furthermore, when a solution exists a special extension (the band extension) is constructed which enjoys a maximum entropy property. A linear fractional parameterization of the set of all extensions is also provided. The techniques used in the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method.