J-inner matrix functions, interpolation and inverse problems for canonical systems, V: The inverse input scattering problem for Wiener class and rational pxq input scattering matrices

The general formulas developed in the fourth paper in this series are applied to solve the inverse input scattering problem for canonical integral systems in the special cases that the input scattering matrix is a p x q matrix valued function in the Wiener class (and the associated pairs are homogeneous). These formulas are then further specialized to the rational case. When p = q, these formulas are connected to the earlier results of Alpay-Gohberg and Gohberg-Kaashoek-Sakhnovich, who studied inverse problems for a related system of differential equations.