The Composition Representation of Time-Varying Systems with Banach-Space-Valued Distributional Signals

This paper studies a linear continuous input-output system, which is in general time varying and whose signals are Banach-space-valued distributions. Such systems can be characterized by two types of composition. The first, which we call "composition
\cdot
," is based on Schwartz's kernel theorem and provides an explicit representation for every continuous linear mapping of smooth functions into distributions. The second, which we refer to as "composition
\circ
," was introduced by Cristescu and Marinescu. In contrast to composition
\cdot
, not all continuous linear mappings of smooth functions into distributions can be represented by composition
\circ
. Composition
\circ
, when it does exist, has the virtue that it can be applied to certain singular distributions. The present work is aimed at this gap between composition
\cdot
and composition
\circ
. A method is developed for extending composition
cdot
onto singular Banach-space-valued distributions in such a fashion that the resulting formula has precisely the form of composition
\circ
. Thus under suitable restrictions on the distributions being composed, composition
\circ
becomes an extension of composition
\cdot
. The fact that the distributional signals are Banach-space valued leads to a variety of complications, all of which are surmounted.