CARATHEODORY-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 necessar y and sufficient conditions for the existence of extensions in terms o f Toeplitz and Hankel operators on Besicovitch spaces and Lebesgue spa ces. 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 als o provided; The techniques used in the proofs include factorizations o f matrix valued almost periodic functions and a general algebraic sche me called the band method.