Opuscula Math. 36, no. 6 (2016), 695-716
http://dx.doi.org/10.7494/OpMath.2016.36.6.695

Opuscula Mathematica

# Characterizations of rectangular (para)-unitary rational functions

Daniel Alpay
Palle Jorgensen
Izchak Lewkowicz

Abstract. We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) through the realization matrix of Schur stable systems, (ii) the Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters, (iii) through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases.

Keywords: isometry, coisometry, lossless, all-pass, realization, gramians, matrix fraction description, Blaschke-Potapov product.

Mathematics Subject Classification: 20H05, 26C15, 47A48, 47A56, 51F25, 93B20, 94A05, 94A08, 94A11, 94A12.

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• Daniel Alpay
• Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel
• Palle Jorgensen
• Department of Mathematics, 14 MLH, The University of Iowa, Iowa City, IA 52242-1419, USA
• Izchak Lewkowicz
• Department of Electrical Engineering, Ben Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel
• Communicated by P.A. Cojuhari.
• Revised: 2016-06-04.
• Accepted: 2016-06-07.
• Published online: 2016-10-29.