Timeline for Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)
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| Mar 31, 2014 at 15:17 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 30, 2014 at 9:53 | comment | added | fedja | As to homework, the answer is "Yes", if the person took a decent course on Banach algebras recently and "No" otherwise. Our (or, at least, my) memory is short and rusty and our education is patchy, so, Noah, I guess I can kill you with an elementary for me question and, most likely, you can return the favor :-). Let us, hence, assume (unless it is obvious otherwise) that whoever asks a question asks it in good faith and for a good reason ;-). | |
| Mar 30, 2014 at 9:44 | comment | added | fedja | Noam has essentially answered it. All we need to notice is that the trick he mentioned allows one to show that the real part of $(1-z)G(z)$ is positive in the closed unit disk except for the point $z=1$, so $G(z)$ cannot be $0$ or negative real (Except, maybe, at $1$? Nah, it equals $1$ there!). Hence, $\log$ has an analytic brunch in some neighborhood ($\mathbb C\setminus(-\infty,0]$) of the compact $G( \text{Clos}\mathbb D )$, so Wiener's theorem finishes the story in no time. | |
| Mar 29, 2014 at 15:36 | answer | added | esg | timeline score: 2 | |
| Mar 26, 2014 at 12:59 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 19, 2014 at 16:01 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 18, 2014 at 13:02 | comment | added | abx | Waouh, 24 edits now! This must be a world record. | |
| Mar 18, 2014 at 11:50 | history | edited | user48365 |
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| Mar 18, 2014 at 8:22 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 18, 2014 at 7:48 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 18, 2014 at 7:43 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 18, 2014 at 5:12 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 18, 2014 at 5:07 | comment | added | user48365 | You are right, but how to proof the absolutely convergent？ I want to use Faa Di Bruno Formula(en.wikipedia.org/wiki/Faà_di_Bruno's_formula) to get the expression of $q_i$. Then to estimate $\sum\limits_{i = 0}^\infty {\left| {{q_i}} \right|}$ | |
| Mar 18, 2014 at 5:07 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 18, 2014 at 4:52 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 18, 2014 at 4:46 | comment | added | Gerry Myerson | 14 edits by author, in under 24 hours. Hard to hit a moving target. | |
| Mar 18, 2014 at 3:06 | comment | added | Noam D. Elkies | There cannot be zeros in $|z|<1$ or even $|z| \leq 1$ because the hypothesis $p_0 > p_1 > p_2 > \cdots > 0$ implies that in the expansion $$ (1-z) \, G(z) = p_0 - \sum_{j=1}^\infty (p_j - p_{j-1}) \, z^j $$ all the coefficients $p_j - p_{j-1}$ are positive and their sum telescopes to $p_0$, so $$ \left| \sum_{j=1}^\infty (p_j - p_{j-1}) \, z^j \right| < p_0$ $$ for all $z \neq 1$ such that $|z| \leq 1$. | |
| Mar 18, 2014 at 2:55 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 18, 2014 at 0:44 | comment | added | user48365 | If z has zeros，lnG(z) has no definition. So we must choose $p_i$ such that lnG(z) has definition. | |
| Mar 17, 2014 at 19:12 | review | Close votes | |||
| Mar 28, 2014 at 14:21 | |||||
| Mar 17, 2014 at 18:51 | comment | added | Alexandre Eremenko | I don't understand the definition of $\log G(z)$ for $|z|<1$. What if your generating function has zeros in the unit disc? | |
| Mar 17, 2014 at 16:12 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 17, 2014 at 15:10 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 17, 2014 at 13:10 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 17, 2014 at 12:02 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 17, 2014 at 11:51 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 17, 2014 at 11:17 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 17, 2014 at 10:48 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 17, 2014 at 10:10 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 17, 2014 at 5:46 | review | First posts | |||
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| Mar 17, 2014 at 5:43 | comment | added | Noah Schweber | Is this homework? | |
| Mar 17, 2014 at 5:41 | history | edited | user48365 | CC BY-SA 3.0 |
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| Mar 17, 2014 at 5:30 | history | asked | user48365 | CC BY-SA 3.0 |