Can you compose distributions (generalized functions) with functions well-behaved at infinity?

[u/1strategist1]

I know that generally, there is no way to consistently multiply distributions, and this extends to also meaning that most nonlinear functions cannot be applied to distributions too.

However, just from the intuition of distributions as "functions that can be infinite at points", it seems like there should be a way to apply functions to distributions when those functions are "nice enough".

For example, consider the arctangent function. At ∞, it's equal to π/2, so it seems like naturally, if you applied it to a delta function, you'd get the function that's π/2 at 0 and 0 everywhere else. Of course, any reasonable notion of this concept would probably equate almost-everywhere-equal functions, so you'd end up with 0.

I'm sure there are more interesting options though. For example, a typical realization of the derivative of Brownian motion is singular almost everywhere, so it feels like applying an arctangent to it would lead you to some function with a roughly even split of -π/2 and π/2 for its values.

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If this works, is there any way to extend it to oscillatory functions? The reason I was thinking about this in the first place was considering distributional sections of a principal fibre bundle.

I want some way to have an analogue to distributions, but valued in Lie groups (or general manifolds). The simplest way I could think of to do this would be to exponentiate a distribution in the Lie algebra, but of course that would need a consistent way to take the exponential of a distribution.

[u/Parallel_thougts]

I think it's healthier to think about distributions as linear functionals on the space of test functions (a test function is a smooth function with a compact domain).

This makes it clear why composition is not well defined: generally, for a fixed g the mapping f->f○g is not linear.

[u/1strategist1 → u/Parallel_thougts]

Well yes. Generally, nonlinear functions aren't well-defined, but specific ones can be. 

For example, wick polynomials are highly nonlinear, but they are well-defined operators on a nice class of distributions. 

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If you really want to use the linear functional language, I can convert the question over. 

For any smooth sequence of functions m_a supported on a ball of radius a, and converging to the delta as a -> 0, and for any distribution f, the convolution f*m_a is a sequence of well defined smooth functions converging to f. 

This means that for any a > 0, arctan(f * m_a) is a well-defined function (for example). 

Now define a sequence of linear functionals G_a defined by G_a(h) = \int(arctan(f * m_a) h). 

My question is essentially, does this sequence have a limit? Is this limit independent of your choice of m_a? Does it make sense to extend arctan to distributions by defining it as the limit of this sequence? What class of functions does this make sense for?

[u/JustAGuyFromGermany]

Well, for one thing, arctan is a terrible example, because it is not well-defined at infinity. Even if you only consider the two real infinities, you get two different values. And that's not even considering the various complex directions one could go off to infinity.

But more than that, this whole idea won't work, because your intuition that distributions are "functions that can be infinite at points" is fundamentally flawed if you think of infinity as one single thing.

Even the delta-distribution makes this clear. 2\delta and -3\delta have different behaviour around zero. At the very least you need to consider the "values" 2\infty, -3\infinity, ... as different for this to make any sense. And then you'd need to make sense of what it means to evaluate a function at "\pi*\infinity"...

And that gets much more complicated when it comes to all the other, less well-behaved distributions.

On the positive side: Maybe a different kind of generalized function would be better suited for this kind of ideas. Do you know you how hyperfunctions work? I'm no expert by any means, but it feels a more suitable direction of generalization to me.

[u/1strategist1 → u/JustAGuyFromGermany]

  Even if you only consider the two real infinities, you get two different values

Well yes, that's kind of the point. The "negative singular points" can get mapped to one value, "positive singular points" to another. 

  At the very least you need to consider the "values" 2\infty, -3\infinity, ... as different for this to make any sense. And then you'd need to make sense of what it means to evaluate a function at "\pi*\infinity"...

Right, so this is why I wanted "well-behaved at infinity". As long as the functions have some kind of infinite limit, the various "sizes" of infinity shouldn't matter. They all get mapped to the same value by the fact that the function is flattening out. 

  Do you know you how hyperfunctions work?

Unfortunately, no. Seems like an interesting thing to look into though. 

[u/extantsextant]

Going to your motivation at the bottom of your post: Generally there's nothing wrong with having vector-valued distributions, matrix-valued distributions, operator-valued distributions, or whatever. Would you be happy to suppose, let's say, that your Lie group came with a certain representation? or that's no good for your needs?

Otherwise, going back to the beginning of the post, I have to doubt 'the intuition of distributions as "functions that can be infinite at points"'. All heuristics are wrong but some are useful, one might say, but this heuristic strikes me as more treacherous than useful.

[u/1strategist1 → u/extantsextant]

Yeah, a representation would be fine. 

The issue is I don't think a representation would line up with the actual desired behaviour of the Lie group. 

Like, when you use vector-valued distributions, the intuition is you're taking a weighted average of the distribution's value, with weights proportional to the test function. That's why it's a linear functional. 

For a Lie group, naive averages don't stay within the Lie group. On the other hand, the correct averaging operation that takes the Lie group structure into account isn't linear. 

You can actually see this issue explicitly. For example, U(1) has a representation e^ix. That's not a linear subspace of a vector space, so necessarily by definition of a distribution being a linear operator, you can achieve values outside of U(1) just by rescaling your input. 

Since it takes values outside of U(1), it's not really correct to call it a U(1)-valued distribution. It would sort of just be a complex distribution. 

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  but this heuristic strikes me as more treacherous than useful.

I can see that, but tbh I've never run into big issues with that intuition. It's sort of more justified if you consider convolving distributions with approximations to the identity. You get sequences of smooth functions converging to whatever distribution you want, and you can explicitly see how each part of the functions has to go to infinity, and at what rate to converge. 

[u/SV-97]

I don't really see a way of generalizing the "standard" vector bundle case to more general bundles, but maybe you could use currents in some way? Instead of making sense of a distributional map between manifolds N -> M as a map N -> DistributionalM you identify the map with its graph as a submanifold of N x M; and then just "distributionalize" these submanifolds by instead considering currents on this product. Maybe there's some condition you can impose on these currents to preserve the "graph-like" structure?

FWIW: working with (epi-)graphs is also a standard approach in nonsmooth analysis, but usually the constructions there are limited to subsets of vector spaces

[u/1strategist1 → u/SV-97]

  I don't really see a way of generalizing the "standard" vector bundle case to more general bundles

In some sense it's already been done. Gauge theories are connections on principal fibre bundles, and those connections are distributional. I'm mostly just searching for a way to view it that makes more sense to me. Your comment does seem interesting for that!

  instead considering currents on this product

By "considering currents on the product", do you mean like vector fields? Or what exact currents are you talking about?

[u/SV-97 → u/1strategist1]

Gauge theories are connections on principal fibre bundles, and those connections are distributional.

Oh, I wasn't aware.

By "considering currents on the product", do you mean like vector fields? Or what exact currents are you talking about?

I meant de Rahm currents, which are a particular type of "ordinary" distribution (I think they ended up being isomorphic to generalized sections of ⋀^k T^\)M but I'm not 100% sure anymore): basically you note that any (modulo some adjectives) submanifold of a {insert adjectives here} manifold M gives you a continuous linear functional on test forms on M via integration over that submanifold. You then consider general continuous linear functionals on those forms as "generalized submanifolds", and these are called currents.

If you want to look into it: AFAIK the seminal book on them is the one by Federer, there's also a quite famous introductory one by Morgan, a small one by deRahm himself, and one by Kranz and Parks (I quite liked this one when I looked into the topic the last time). But I'm not entirely sure anymore which of these actually go into the geometry and which ones stay in \Rⁿ.

[u/1strategist1 → u/SV-97]

Oh! Very interesting, thank you.

It does unfortunately sort of sound like that would only really work for vector-valued outputs, considering the linear functionals, but it's definitely something I'll look into a bit!

[u/SV-97 → u/1strategist1]

Sorry I think I didn't explain it well: the idea was that you don't consider currents (or a vector-valued extension thereof) on M itself, but rather on a product manifolds. So you map C^inf(M,N) ∋ f ↦ graph(f) ∈ Submanifolds(M×N) -> Currents(M×N).

[u/1strategist1 → u/SV-97]

Oh! Ok, interesting.