## Abstract

A fundamental object of study in both operator theory and system theory is a discrete-time conservative system (variously also referred to as a unitary system or unitary colligation). In this paper we introduce three equivalent multidimensional analogues of a unitary system where the "time axis" ℤ ^{d}d>1 is multidimensional. These multidimensional formalisms are associated with the names of Roesser Fornasini and Marchesini and Kalyuzhniy-Verbovetzky. We indicate explicitly how these three formalisms generate the same behaviors. In addition we show how the initial-value problem (including the possibility of "initial conditions at infinity") can be solved for such systems with respect to an arbitrary shift-invariant sublattice as the analogue of the positive-time axis. Some of our results are new even for the d=1 case.

Original language | English |
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Pages (from-to) | 133-198 |

Number of pages | 66 |

Journal | Multidimensional Systems and Signal Processing |

Volume | 16 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2005 |

## Keywords

- Conservative multidimensional discrete-time system
- Energy balance relation
- Initial value problem boundary condition at infinity
- Shift-invariant sublattice

## ASJC Scopus subject areas

- Software
- Signal Processing
- Information Systems
- Hardware and Architecture
- Computer Science Applications
- Artificial Intelligence
- Applied Mathematics