Recent representation results for linear system maps: a short survey.

*(English)*Zbl 1128.93027Summary: We give an expression for the most general input-output map associated with the members of a certain important large family of multidimensional linear shift-invariant systems with bounded Lebesgue-measurable inputs. The expression given is an iterated function-space limit of a convolution. We also give a necessary and sufficient condition under which the limit can be written as a convolution with an integrable impulse-response function. A key role is played by a certain family of weighting operators. It is observed that for the large family of inputs and maps addressed, the Dirac impulse-response concept is in fact not the key concept concerning the representation of \(H\), and that instead the input-output properties of \(H\) are determined, in general, by a certain type of family of responses. Some related material concerning other results, engineering education, and discrete-space systems, is also given.

##### MSC:

93C35 | Multivariable systems, multidimensional control systems |

93C65 | Discrete event control/observation systems |

93C05 | Linear systems in control theory |

##### Keywords:

linear systems; multidimensional systems; impulse responses; shift-invariant systems; bounded measurable inputs; discrete-space systems
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\textit{I. W. Sandberg}, Int. J. Circuit Theory Appl. 35, No. 5--6, 497--514 (2007; Zbl 1128.93027)

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