Distributions & test functions in specific applications

[jasonRF]

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/)

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

 

[micromass]

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/)

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 $\langle \delta, f \rangle = f(0)$.

That is most definitely allowed. As long as the origin has nice properties, then you can do this. The trick here is that the evaluation of the delta function is a local property. So if we are nice and smooth in the origin, then it doesn't matter that there are discontinuities far away.

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.

This seems a bit trickier. But there are definitely ways to "eliminate" discontinuities and poles. This is by no means an elegant theory, but it exists.

Finally, if all you care about is a delta function, then you should look into Riemann-Stieltjes (or Lebesgue-Stieltjes) integration. That deals with the delta function very nicely, and it doesn't need the more general theory of distributions.

 

[jasonRF]

Thanks micromass. I haven't run into trouble doing this kind of thing in the past, but was curious what more informed folks thought.

As an engineer I usually use things like delta functions and their derivatives as an approximation to make most of the math easier. For example my system has some pulsed signals that are much much shorter than the time scales of the other signals, so I use a delta function to be able to quickly do the math and understand approximately how the system works. Another example: when one signal is very narrow band relative to the other signals, I will use a delta function in the frequency domain to describe it. When the approximation stops making the math easier (trying to compute the energy of the pulse = $\int dt \, \delta(t) \delta(t)$ which is nonsense), I switch back to a higher fidelity model that is messier (must keep track of leading and trailing edges of pulses, etc.) but well behaved.

I know the basic idea of Riemann-Stieltjes integration, as it comes up in probability theory to allow single proofs to be applicable for both continuous and discrete random variables. When modelling signals, currents on small antennas, etc. it is actually easier for me to think about delta functions since I can essentially think of them as functions, or approximations to the physical reality. I'm not sure how to build my models (or derive a Green's function) in a way that revolves around the assumption that much later in the modeling process I will be using Stieltjes integration.

Thanks

jason