limit of distribution
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Please post an actual problem so that we can limit our discussion around that.
Dr.Peterson
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How are you defining delta(x) here?
How do you "know" the limit is zero? Did you read it somewhere, or it that a guess?
Dr.Peterson
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Dirac delta
I wondered if you might mean the Dirac delta; but "delta" is used in many different ways! Context is essential, especially as you get farther into advanced subjects. For that matter, "distribution" means a lot of different things. Putting "Dirac delta" into your title would have helped set the context.
There's still a lot more context you could share, for the benefit of those who might not have the same textbook; notation at this level is not universal. I'm going to have to take some time looking into this, in case no one else looking at your question is more current on it than I am.
Meanwhile, can you tell us more about what the solution said, and show the work you have done? Anything you can add will make it easier for others to help you.
I wondered if you might mean the Dirac delta; but "delta" is used in many different ways! Context is essential, especially as you get farther into advanced subjects. For that matter, "distribution" means a lot of different things. Putting "Dirac delta" into your title would have helped set the context.
There's still a lot more context you could share, for the benefit of those who might not have the same textbook; notation at this level is not universal. I'm going to have to take some time looking into this, in case no one else looking at your question is more current on it than I am.
Meanwhile, can you tell us more about what the solution said, and show the work you have done? Anything you can add will make it easier for others to help you.
the context of my question is the following
https://en.wikipedia.org/wiki/Distribution_(mathematics)
Dr.Peterson
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Yes, I knew that much from what you showed of the definition. I was hoping for a little more specific context of what you have learned about it, and any pieces of notation that might differ from what we can both read there. Did you have any specific theorems, or any similar examples?
Since you haven't shown your work as I asked (and as we always ask), let's start working through it. Can you tell me what δ(1/h) is? Then we can think about the question, \(\displaystyle \displaystyle\lim_{h\to \infty} h \delta\left(\frac{1}{h}\right)\).
yeah, it is phi(1/h)
where phi is a test function compactly supported, which means that above an interval (a,b) it becomes exactly zero.
so for example the limit as h goes to infinity of delta(h), since it is phi(h) would be 0.
but here I have phi(1/h) which shouldn't be 0, or maybe it can be, but then it is multiplied by h (so infinity),so I don't know how to proceed
Dr.Peterson
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How can δ(1/h) be φ(1/h)? I don't think you mean what you are saying. The Dirac delta function is not the same thing as any of the test functions used to define it.
I want to know what the question means by δ(1/h)! What does that notation mean in your textbook? In Wikipedia, I see the notation \(\displaystyle \langle\delta,\phi\rangle\); I don't see \(\displaystyle \delta(x)\) or \(\displaystyle \delta(f)\)or \(\displaystyle \delta(\phi)\).
Keep in mind that I don't think I've worked with this in 40 years, so I need you (or someone else!) to help me out. Things are coming back to me, but variations in notation don't help. I need to see what you've been taught. If I were working with you face to face, I would have grabbed your book and skimmed the chapter to remind myself of what I've forgotten, and to see how your book says things. Is there a definition of the notation? Is there an example of how to work out a limit like this?
Now, if it means what I think it means, the answer may be almost trivial. But I don't want to say that without being sure I know what the question is asking.
Dr.Peterson
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Here's my main difficulty: the delta function, as a distribution, is a functional on the set of test functions, D(R). So the only way I can interpret δ(1/h) is to take 1/h as the function f(h) = 1/h; but this function does not have compact support, and is not infinitely differentiable, so it is not in D(R), the domain of delta.
We could take it to mean the Dirac delta thought of as if it were a function on R, so that for h ≠ 0, δ(1/h) = 0. Then h*0 = 0, so the limit is zero. But this doesn't make use of the definition you have been given, right? Or were you taught an alternative notation that you haven't shown us?
This is why we really need to see how the definitions have been presented to you, and how they handle questions like this. Possibly we need to see an image of a page of your textbook.