Abstract
We study inverse problems associated to first-order discrete systems in the rational case. We show in particular that every rational function strictly positive on the unit circle is the spectral function of such a system. Formulas for the coefficients of the system are given in terms of realizations of the spectral function or in terms of a realization of a spectral factor. The inverse problems associated to the scattering function and to the reflection coefficient function are also studied. An important role in the arguments is played by the state space method. We obtain formulas which are very similar to the formulas we have obtained earlier in the continuous case in our study of inverse problems associated to canonical differential expressions.
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Alpay, D., Gohberg, I. (2007). Inverse Problems for First-Order Discrete Systems. In: Ball, J.A., Eidelman, Y., Helton, J.W., Olshevsky, V., Rovnyak, J. (eds) Recent Advances in Matrix and Operator Theory. Operator Theory: Advances and Applications, vol 179. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8539-2_1
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