A characterization of bounded-input bounded-output stability for linear time-invariant systems with distributional inputs

We consider linear time-invariant operators defined on the space of distributions with left-bounded support. We argue that in this setting the convolution operators constitute the most natural choice of objects for constructing a linear system theory based on the concept of impulse response. We extend the classical notion of bounded-input bounded-output stability to distributional convolution operators and determine precise conditions under which systems characterized by such maps are stable. A variety of expressions for the ''gain'' of a stable system is derived. We show that every stable system has a natural threefold decomposition based on the classical decomposition of functions of bounded variation. Our analysis involves certain extensions of the Banach spaces L(p) in the space of distributions.