Products and compositions with the Dirac delta function

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Abstract
The need to define pointwise products and compositions with distributions is pointed out in the context of the problems of renormalisation, junction conditions and curved shock waves. Earlier definitions are briefly reviewed, and new definitions are proposed using non-standard analysis. Basic properties are established, and some products and compositions with the delta distribution are explicitly evaluated. With these definitions, the domain of validity of the nonlinear differential equations of classical field theory can be extended to include Rankine-Hugoniot equations are derived from the Euler equations. An immediate application to quantum field theory is pointed out.

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