Timeline for Stone-Weierstrass without the "subalgebra" condition
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 15, 2023 at 7:45 | comment | added | terceira | $$\sum \left(1-\left|\frac{1-z_p}{1+z_p}\right|\right)$$ diverges. | |
| May 15, 2023 at 7:38 | comment | added | terceira | It is possible to give a direct proof of a very general criterion for the completeness of sequences in $c_0$ which includes your situation and only uses Hahn-Banach and a well known condition for a sequence in the open right plane to be the zero set of a bounded holomorphic function there. If $(x_n)$ is a positive sequence in the unit ball of $c_0$ which separates the indices (in your case $(\frac 1 n)$), then $({x_n}^{z_p})$ ($p=1,2,\dots$) is complete whenever | |
| May 5, 2023 at 16:38 | history | became hot network question | |||
| May 5, 2023 at 14:05 | answer | added | Pietro Majer | timeline score: 4 | |
| May 5, 2023 at 13:59 | answer | added | Sean Eberhard | timeline score: 8 | |
| May 5, 2023 at 12:52 | comment | added | Sean Eberhard | Since $\alpha$ is irrational, every $p_t(n) = 1/n^t$ for $0 < t \le 1$ can be approximated by some $p_{\{m\alpha\}}$. Therefore it is the same to ask about the span of $\{p_t : 0 < t \le 1\}$. | |
| May 5, 2023 at 12:43 | comment | added | Nik Weaver | Eh, I am a little sick right now, so maybe not thinking clearly. You are right. | |
| May 5, 2023 at 12:38 | comment | added | Ewrt Wert | But the exponent is constant for each function | |
| May 5, 2023 at 12:27 | comment | converted from answer | Ewrt Wert | (Add to comment: I read too fast, but as your functions separate the points, the only question is if norm closure of span of $p^{\{mα\}}$ forms algebra -> this might be questioned) (How do you write $e^{-n}$ or only $n^{-2}$ approximately as linear span of your functions $1/n^{u}$ with $0 < u < 1$?) | |
| May 5, 2023 at 11:23 | comment | converted from answer | Ewrt Wert | (Add to comment: even $p^\alpha$ alone should separate the points and generate your algebra, essentially, modulo unit) | |
| May 5, 2023 at 9:04 | comment | added | F J | Yes, the sets of functions should be $\{p_{m\alpha},m\geq 1\}$ and $\{p_{\{m\alpha\}},m\geq 1\}$. | |
| May 5, 2023 at 8:47 | comment | added | Goldstern | When you write $\{p_{m\alpha}\},m\geq 1$, you mean $\{p_{m\alpha}\mid m\geq 1\}, right?$ | |
| May 5, 2023 at 8:37 | history | asked | F J | CC BY-SA 4.0 |