Translation Varying Linear Operators
Reuven Meidan · SIAM Journal on Applied Mathematics · 1972
Translation varying linear operators which are continuous from $\mathcal{D}( {X^m } )$ into $\mathcal{D}'( {Y^n } )$ are investigated. Certain postulates motivated by physical considerations are imposed on the operators. One is the postulate of regularity according to which the output of the operator is assumed to be a continuous function (where the input is an infinite continuously differentiable function of compact support). This assumption is justifiable from a physical viewpoint and is weaker than the assumption of translation invariance. It is shown that some of the essential properties of translation invariant operators are associated with the assumption of regularity. In particular, a modified impulse response and a scalar product representation are developed for the translation varying regular operators. The other postulate considered is passivity in the framework of the scattering formalism and in the weak sense. Characterizations of passive kernel operators are pursued and an isomorphism between contractions in $L^2 $ and passive kernel operators is established. This provides a kernel theorem and a scalar product representation for bounded operators in $L^2 $.