Causality and a Theorem of Paley and Wiener

Abstract

AbstractIn this chapter we turn our focus back to causal operators. In Chap. 10.1007/978-3-030-89397-2_5 we found out that material laws provide a class of causal and autonomous bounded operators. In this chapter we will present another proof of this fact, which rests on a result which characterises functions in $$L_2(\mathbb {R};H)$$ L 2 ( ℝ ; H ) with support contained in the non-negative reals; the celebrated Theorem of Paley and Wiener. With the help of this theorem, which is interesting in its own right, the proof of causality for material laws becomes very easy. At a first glance it seems that holomorphy of a material law is a rather strong assumption. In the second part of this chapter, however, we shall see that in designing autonomous and causal solution operators, there is no way of circumventing holomorphy.