Continuous linear shift invariant systems on tempered distributions: convolutors and convolution

Abstract This paper characterizes the Continuous, Linear and Shift Invariant Systems on Tempered Distributions. The needed classical notions on Distributions, and on Tempered Distributions, are reminded. The delicate and unusual subjects of Convolutors, and of their Convolution with Tempered Distributions, are made accessible, and easy to handle, through autonomous elementary definitions: equivalence with Schwartz’s Definitions is proven. Convolutors describe the family of the Continuous Linear, and Shift Invariant Systems on Tempered Distributions; Convolution describes their behaviour.