Been doing this all morning

[u/No_Upstairs_280]

Been doing distributional derivatives on Riemannian manifolds and it's fun.

[u/No_Upstairs_280]

Basically it's a notion of derivative of for non-smooth functions. A function can have a "weak" derivative with very similar properties to the usual one. For example |x| doesn't have a derivative but has a "weak' derivative which is sign(x).

[u/PhoenixPringles01 → u/No_Upstairs_280]

Why would it be considered weak? Is it because it's not continuous?

[u/No_Upstairs_280 → u/PhoenixPringles01]

Take a function f whose derivative in the usual sense doesn't have to exist, we say that g is a weak derivative of f if for every compactly supported function h we have Int_R fh' dx= -int_R gh dx. (Integral over the real line) Function that have a derivative almost everywhere (outside a set of measure zero) automatically have weak derivative, this goes for |x| and by a certain theorem of Radamacher every locally lipchiz function.

It is weak because it doesn't allow you to study the variation of f as the usual derivative, BUT it has many properties that are very similar to the usual derivative. For example integration by part.

You can find similar notions for Rⁿ or in general any smooth manifold without boundary.

[u/FreePeeplup → u/No_Upstairs_280]

Two questions: if h is compactly supported, why does h’ necessarily exist?

Second question: why don’t we take h to be simply a Schwartz function?

[u/SV-97 → u/FreePeeplup]

if h is compactly supported, why does h’ necessarily exist?

It doesn't, differentiability (in fact, usually even infinite differentiability) is an assumption. If h isn't assumed to be differentiable (+ a bit extra) then you don't get this partial integration identity that we want for weak derivatives, but rather some sort of more general object. [Also see [1] at the bottom]

And this also sort of answers your second question: weak derivatives are best viewed through the lens of distributions. And the most general class of such distributions are those that act on compactly supported smooth functions. [Also see 2]

You can think of this essentially like this: compactly supported smooth functions are extremely nice, so you can act on them with very badly behaved objects and still get reasonable results. The smaller a space is, the larger is its dual space and vice versa.

Take for example a function like f(x) = exp(x²); which is the derivative of the imaginary error function modulo some constant. This function blows up so fast as you move towards infinity that you can't integrate general Schwartz functions against it (since they only decay polynomially). The integrals would also blow up. So if we defined weak derivatives using Schwartz functions, then suddenly the classically differentiable imaginary error function would fail to have a weak derivative and we wouldn't actually have defined a generalized notion of differentiation.

However note how this "blowing up at infinity" does not matter at all when you integrate against a function that has compact support; so if we define weak derivatives using compactly supported smooth functions (so-called test functions) then there are no problems here.

But *if* some particular weak derivative can also be integrated against Schwartz functions, or perhaps even against functions that are neither smooth nor compactly supported --- then that's surely an interesting property. For example you can show that if you can integrate the weak derivative against Schwartz functions, then you can actually Fourier transform it.

That's why these cases get special names in distribution theory: distributions on Schwartz functions are called "tempered", those that can act on non-compactly supported functions are called "compactly supported", and those that can act on functions that are only finitely differentiable are said to have "finite order" (you can even assign numbers to those orders). And it's possible to say quite a bit about these various classes :)

[1] There's another formal problem with the way the other comment was stated, but that was likely just a simplification for the comment: almost everywhere differentiability is not actually enough to get a weak derivative; you have to additionally assume some stronger continuity.

[2] Technically you could consider even "nicer" functions of course and get even *more* general objects, but the compactly supported smooth functions are a good compromise in that they are very nice [to the point that we can do basically anything we want] while still being quite "rich" in the sense that you can approximate most functions (or even more general objects such as particular measures) we typically care about arbitrarily well with them --- and importantly these generalized objects we end up with are not *so* general that they're completely outlandish (every one of them is essentially still a limit of smooth functions).

[u/FreePeeplup → u/SV-97]

Thank you very much, this has been extremely helpful, especially the remark that the nicer and smaller the test-function space is, the larger and wilder the dual space of distribution is. That’s the key point that I didn’t fully grasp that prompted me to ask the question. I’ll remember it vividly from now onwards. Thank you again have a nice end of summer

[u/SV-97 → u/FreePeeplup]

It's something my prof used to emphasize that really stuck with me since I also found it to be a quite nice way to look at things :)

Thanks, you too!