Timeline for Riemann rearrangement theorem for $L^1$ functions
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Apr 29, 2021 at 22:42 | vote | accept | Nate River | ||
| Apr 29, 2021 at 7:10 | answer | added | mlk | timeline score: 8 | |
| Apr 29, 2021 at 3:47 | comment | added | Eric | Maybe, try ordering the $c_n$ so that the numbers of the same sign are decreasing in magnitude and the sum of $\delta_i c_i$ converges to the integral of $f$. Then, if $c_1$ is negative let $A_1$ be the largest measure $\delta_1$ values of $f$. Subtract this out of those values of $f$. If $c_1$ were negative, then choose the smallest values. Keep doing this, repeatedly decreasing the largest values and increasing the smallest values. Not sure how to prove this works - Maybe you can get bounds on how fast the $L_1$ norm decreases. | |
| Apr 28, 2021 at 20:45 | history | edited | Nate River | CC BY-SA 4.0 |
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| Apr 28, 2021 at 20:43 | comment | added | Nate River | Oh that isn’t true at all. You’re right, I will think about what needs to be added. | |
| Apr 28, 2021 at 20:38 | comment | added | Nate River | @WillieWong hmm I think the condition that $\sum c_n$ converges conditionally and $\delta_n \to 0$ (in particular, being eventually less than 1) should ensure the conditional convergence of $c_n \delta_n$. | |
| Apr 28, 2021 at 19:20 | comment | added | Willie Wong | You'd need at least that $c_n \delta_n$ is also conditionally convergent, not just with divergent absolute values. | |
| Apr 28, 2021 at 19:16 | comment | added | Willie Wong | As stated I don't think it would work. For example, let $c_n$ be the alternating harmonic sequence. Let $\delta_n = \frac{1}{\ln(n+3)}$ if $n$ is odd, and $\frac{1}{n}$ if $n$ is even. Then you cannot use this to approximate any $f$ whose negative part has mass greater than 10. | |
| Apr 28, 2021 at 8:45 | history | edited | Nate River | CC BY-SA 4.0 |
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| Apr 28, 2021 at 8:44 | comment | added | Nate River | Ah silly me, I forgot to require that $d_n \to 0$ @Leo Moos. And thanks mlk that’s a good suggestion too! | |
| Apr 28, 2021 at 8:31 | comment | added | mlk | If you pick $|c_n| = \delta_n = 1/n$, then you can estimate $\int | \sum c_{\gamma(n)} 1_{A_{\gamma(n)}} | dx \leq \sum |c_n| \delta_n = \sum \frac{1}{n^2} < \infty$, so to approximate arbitrary functions, you might need to also require that $\sum |c_n| \delta_n = \infty$. | |
| Apr 28, 2021 at 8:10 | comment | added | Leo Moos | I don't think this works when $\delta_n = 1$. | |
| Apr 28, 2021 at 7:19 | comment | added | Nate River | $A_n$ are chosen freely, so there should be no constraints. | |
| Apr 28, 2021 at 7:10 | comment | added | Geoffrey Irving | Are you missing some constraints on $A_n$? As is you can take them to all be the same set, or to be disjoint from the support of $f$. | |
| Apr 28, 2021 at 6:15 | history | edited | Nate River | CC BY-SA 4.0 |
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| Apr 28, 2021 at 6:09 | history | asked | Nate River | CC BY-SA 4.0 |