مرکز منطقه ای اطلاع رساني علوم و فناوري - The Composition Representation of Time-Varying Systems with Banach-Space-Valued Distributional Signals

Abstract :

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.

Keywords :

Banach spaces; Distributed systems, linear time-varying; Distribution theory; Function analytic methods; Linear systems, time-varying continuous-time; Airborne radar; Artificial satellites; Books; Bridges; Communication equipment; Design engineering; Helium; Laboratories; Telephony; Time varying systems;