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Plenary lecture I: the convolution and impulse response scandals

It is a widely-held belief that the main textbook conclusions concerning continuous-time linear systems obtained using Dirac delta-function arguments can be shown to be valid using the mathematical theory of distributions. But this belief is unwarranted. For example, in a recent study of multidimensional input-output maps representing linear shift-invariant systems that take a set of continuous-space signals into itself, it was shown that the family contains maps - even causal continuous maps - whose impulse response is the zero function, but which take certain inputs into nonzero outputs. In this connection, we give recent results concerning the representation of the input-output map associated with the members of a certain important large family of multidimensional linear systems. We also give necessary and sufficient conditions under which the representation can be written as a convolution, and we relate this to the concepts of an impulse response and a q-response limit, and to the flawed concept of the Dirac function.