Timeline for Does this sequence span $L^2$?
Current License: CC BY-SA 2.5
9 events
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| Jun 11, 2010 at 17:33 | comment | added | Will Jagy | Also, Wadim, England-USA in the (soccer) World Cup is on TV in about 25 hours from this post. You can cheer for either team, both, or neither, but you are required to watch. There will be a quiz. | |
| Jun 11, 2010 at 17:24 | comment | added | Will Jagy | Hi, Wadim. I'm going to try to see what I can make of the fact that the Dirac delta function at $0$ is orthogonal to all the given functions. In truth, I do not know the related machinery that well. | |
| Jun 11, 2010 at 10:33 | comment | added | Wadim Zudilin | Thank you, Philipp! I won't remove my answer any way, the people are free to downvote. Something which I do understand about MO now is that it's better not to attempt solving problems (especially which do not directly refer to my area of expertise) but answering questions. The difference between "problems" and "questions" is huge, and the MO community welcomes the latter much better. The former are of interest to a very small group (with very rare exceptions). I believe I have many other problems to solve, and this will be better spending of time for me. But reading MO posts remains fun. | |
| Jun 11, 2010 at 10:04 | comment | added | Philipp | Wadim, I did not want to criticize your argument but understand it better. This was not meant as an attack or anything, so no reason to be angry. You can also see the asking of questions positively: I asked because I found your answer interesting! | |
| Jun 11, 2010 at 9:55 | comment | added | Matthew Daws | An obvious point... but the original question certainly does not ask that (f_n) forms an orthonormal basis: only if the linear span is dense. | |
| Jun 11, 2010 at 9:48 | comment | added | Guy Katriel | The above argument would seem to imply, for example, that the sequence $x,x^2,x^3,...$ does not span $L^2[0,1]$ (since all these functions vanish at $0$) - but in fact it does, by simple arguments. The problem is, as pointed out by Philipp, the implication from $L^2$ to pointwise convergence. | |
| Jun 11, 2010 at 9:46 | comment | added | Wadim Zudilin | Well, assuming that the derivative of a function $f(x)$ exists and is square integrable implies the absolute convergence of the Fourier series... This is specific for standard bases however. But if we pose the condition on $f(x)$ to be analytic on the whole $\mathbb R$, then everything will be absolutely and uniformly convergent (example $e^{-x^2}$). I don't know why do I answer the question; after all these I find it boring. The other answers give hints and nobody complains, while I am immediately asked to give more details. It's not a solution but a hint! | |
| Jun 11, 2010 at 9:06 | comment | added | Philipp | How does the $L^2$-convergence of the sum imply the pointwise convergence? | |
| Jun 11, 2010 at 8:57 | history | answered | Wadim Zudilin | CC BY-SA 2.5 |