Timeline for Does this sequence span $L^2$?
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| Jun 13, 2010 at 16:14 | comment | added | Jeff Schenker | I hesitate to make the conjecture because when $\sum n \frac{a_n}{(1+a_n)^2} <\infty$, we get more than that the span of your sequence has dense range. In fact, the span of the larger collection $f_{n,k}=x^k e^{-a_n x}$ where $1\le k\le n<\infty$ fails to be dense. However that could focus your problem a bit: if you can show that $x^k e^{-a_n x}$ for $k<n$ is in the closure of the span of your functions then you should be able to conclude that this condition is necessary and sufficient. | |
| Jun 13, 2010 at 11:28 | comment | added | Guy Katriel | Thank you Jeff, that's very interesting. So one could conjecture that $\sum_{n=1}^\infty n \frac{a_n}{(1+a_n)^2} =\infty$ is a necessary and sufficient condition for the sequence $f_n$ as you defined to span $L^2[0,\infty)$. The that $a_n$ is constant is the most famous one, with the Laguerre functions used to solve the Hydrogen atom. So you gave a very nice proof of the completeness of the Lagueree functions. | |
| Jun 13, 2010 at 10:13 | history | answered | Jeff Schenker | CC BY-SA 2.5 |