I have a question inspired by a recent thread that I did not want to hijack (https://www.physicsforums.com/threads/distributions-on-non-test-functions.831144/)(adsbygoogle = window.adsbygoogle || []).push({});

I realize that weaker requirements on the space of test functions results in a more restricted set of distributions. For example, if we simply required the test functions to be continuous, then we can have a delta distribution but not the derivative of the delta. This prevents us from trying to differentiate a test function at a point where it is not differentiable.

However, in applications (where we often ignore rigor for the most part) we sometimes have a specific function we care about instead of an entire space of test functions. For example, we may have a delta distribution ##\delta## acting on a specific discontinuous function ##f##. As long as the discontinuity is not at the point of the delta function (say at the origin) we simply ignore the discontinuity and set ##\langle \delta, f \rangle = f(0) ##.

Is there a way to legitimately do this kind of analysis, or is this playing with fire?

EDIT: or can we actually define a space of test functions that includes discontinuous functions, and define distributions on this space that includes the delta? It seems like there would have to be many additional cumbersome "caveats" included when defining all of the various opertations we care about - translation, differentiation, etc. to ensure we never evaluate a test function at a point of discontinuity. This seems like a bad (and in no way elegant) way to go, but just wondering.

jason

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# Distributions & test functions in specific applications

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