Timeline for How to verify the weak convergence?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| May 12, 2011 at 11:25 | comment | added | bib | Maybe I am wrong, the above example by Yemon Choi seems to work again. | |
| May 12, 2011 at 11:15 | comment | added | bib | Nice... It seems we then have the weak convergence of the arithmetical means, isn't it? | |
| May 11, 2011 at 21:16 | comment | added | Bill Johnson | The needed "something extra" is a condition that guarantees that the sequence has weakly compact closure in $L^1$, such as no subsequence of $f_n$ is equivalent to the unit vector basis for $\ell^1$. | |
| May 11, 2011 at 7:34 | comment | added | bib | Zen Harper, many thanks to you too. I see that the general argument does not work, so I need to estimate what I have. | |
| May 11, 2011 at 7:29 | comment | added | Zen Harper | In the other direction, asking what extra assumptions on $(f_n)$ and the measure will guarantee the answer "yes" seems to me like an interesting question, possibly connected with Tauberian theories. This is exactly the kind of question which can make research interesting, in my opinion, since I doubt there will be any precise necessary/sufficient characterisation from general theory which isn't just tautological. | |
| May 11, 2011 at 7:27 | comment | added | bib | Interference for a step function supported near zero - ok. Thanks. I'll try to understand what this means for my question. | |
| May 11, 2011 at 7:25 | answer | added | Pietro Majer | timeline score: 2 | |
| May 11, 2011 at 7:17 | comment | added | Yemon Choi | @bib: I don't know of a textbook that would cover this precise question, but it may help to think of the following version: try to find a bounded sequence $(f_n)$ in $L^1[-1,1]$ such that $\int_{-1}^1 f_ng \to 0$ for every continuous $g$ but with $\int_0^1 f_n$ not converging to zero. | |
| May 11, 2011 at 7:14 | comment | added | Zen Harper | Well, you really need to be a lot more specific for the problem you have; one nice example is from Fourier Series: take $f_n(x) = e^{inx}$ on the compact interval $[0,2\pi]$ with Lebesgue measure, then the limit is zero for any $g \in L^2$, say (indeed for any $g \in L^1$ by the Riemann-Lebesgue Lemma). I suggest you give a detailed interesting special case which you can't resolve, and then try to see if general theory can be applied. (Actually, most general, abstract theory is originally motivated by interesting special questions - but many textbooks try to hide this!) | |
| May 11, 2011 at 7:12 | comment | added | bib | Can you recommend some reading for homework, please. | |
| May 11, 2011 at 7:08 | comment | added | Yemon Choi | @Zen: by the time I wrote my 2nd comment, I was starting to reconsider... and on reflection you are right about the level. (By coincidence I was thinking about a very similar question some months ago.) | |
| May 11, 2011 at 6:52 | comment | added | bib | Zen Harper: Can you suggest simple extra assumptions when this is true? | |
| May 11, 2011 at 6:49 | comment | added | Zen Harper | Yemon, I agree this is a homework problem (or if not, it should be), but (possibly) at graduate level, so I think you're being a little harsh; it is conceivable that a non-(functional analyst) research mathematician, or even a (non-functional) analyst, might find this a bit tricky. There are many interesting special cases of $f_n$ where the answer is "yes"! | |
| May 11, 2011 at 6:39 | comment | added | bib | I need a generalization of this fact for another pair of spaces, where I want to borrow arguments from this standard case. If agruments like the Hahn-Banach theorem work, this could probably help, but I could not restore the proof. | |
| May 11, 2011 at 6:36 | comment | added | Yemon Choi | The answer, by the way, is "no". Could you say a little more about where you came across this problem, which special cases you have already tried, and so forth? | |
| May 11, 2011 at 6:33 | comment | added | Yemon Choi | mathoverflow.net/faq#whatnot Voting to close. | |
| May 11, 2011 at 6:28 | history | asked | bib | CC BY-SA 3.0 |