In this paper we consider functions in the Hardy space Hp×q2 defined in the unit disc of matrix-valued. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke product, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. We show that the maximum selection principle gives rise to a convergent algorithm. Finally, we discuss the case of real-valued signals.
D. Alpay, F. Colombo, I. Sabadini and T. Qian. Adaptive orthonormal systems for matrix-valued functions. Proceedings of the AMS, vol. 145, number 5 (2017), no. 5, 2089-2106.
American Mathematical Society