Timeline for Are the jumps of a càdlàg function "summable"?
Current License: CC BY-SA 4.0
13 events
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| Aug 2, 2023 at 1:19 | history | edited | Julian Newman | CC BY-SA 4.0 |
added update
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| Aug 2, 2023 at 1:14 | comment | added | Julian Newman | @IosifPinelis I am only defining the sum when an $\mathbf{x}$-nice partition exists; the existence of an $\mathbf{x}$-nice partition is part of my definition of the summability of $\mathbf{x}$. | |
| Aug 2, 2023 at 1:12 | answer | added | Julian Newman | timeline score: 1 | |
| Aug 2, 2023 at 0:44 | comment | added | Julian Newman | @AnthonyQuas Sorry, the limit I wrote probably doesn't exist, but I guess your point is that a construction as in the answer to math.stackexchange.com/questions/10257 is possible for your $g$. | |
| Aug 2, 2023 at 0:35 | comment | added | Julian Newman | @AnthonyQuas Okay, I guess you're defining $\sum_{a \in S,\ a\leq x} g(a)$ as $\lim_{n \to \infty} \sum_{a \in S_n,\ a\leq x} g(a)$ with $S_n=\{\frac{j}{3^n} : 1 \leq j \leq 3^n\}$. Nice idea - it probably works! | |
| Aug 2, 2023 at 0:18 | comment | added | Julian Newman | @AnthonyQuas Forgive the probably stupid question, but how are you defining $\sum_{a \in S} g(a)$ given that [presumably as the whole point of your example] $\sum_{a \in S} |g(a)|=\infty$? | |
| Aug 2, 2023 at 0:04 | comment | added | Julian Newman | @IosifPinelis I'm defining by transfinite recursion: unless I've made a mistake, there will exist a unique $\left( \sum_{\alpha \in J}^{\mathcal{P},\leq} x_\alpha \, : \, \text{initial segment } J \subset S \right)$ fulfilling the three bullet points in my definition. [Note that in the case that you've asked about, taking $\mathcal{P}$ to be the partition into singleton sets is only possible if $\mathcal{D}(f)$ is well-ordered.] | |
| Aug 1, 2023 at 23:54 | comment | added | Iosif Pinelis | Your definition of the sum is quite unclear. In particular, if $\mathcal P$ is the partition into the singleton sets and $J=S=\mathcal D(f)\ni1$, then how is $\sum_{\alpha \in \bigcup(\mathcal{P}(J) \setminus \{I\})}^{\mathcal{P},\leq} x_\alpha$ defined, for $x_t:=f(t)-f(t-)$? | |
| Aug 1, 2023 at 23:47 | comment | added | Anthony Quas | I'm having a bit of a hard time parsing your definitions but here is a construction that might yield a non-summable cadlag function. Let $S$ be the set of triadic rationals, so that any $a\in S$ can be written as $\frac j{3^n}$ for some $j$ and $n$. Define $g(\frac j{3^n})$ to be $1/2^n$ if $j\equiv 1\pmod 3$ and $-1/2^n$ if $j\equiv 2\pmod 3$. Then build a function $f(x)=\sum_{a\in S\,a\le x}g(a)$. | |
| Aug 1, 2023 at 22:58 | history | edited | Julian Newman | CC BY-SA 4.0 |
corrected typo in final display equation
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| Aug 1, 2023 at 22:26 | history | edited | Julian Newman | CC BY-SA 4.0 |
changed "process" to "function" in title
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| Aug 1, 2023 at 21:41 | history | edited | Julian Newman | CC BY-SA 4.0 |
small clarifying improvements
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| Aug 1, 2023 at 21:26 | history | asked | Julian Newman | CC BY-SA 4.0 |