Timeline for An extension of Morera's Theorem
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 11, 2014 at 2:49 | vote | accept | booksee | ||
| Sep 29, 2013 at 22:02 | comment | added | booksee | A jointly function is the point sweeping away my doubts. Thanks a lot ! | |
| Sep 29, 2013 at 19:55 | comment | added | Sean Eberhard | @booksee Let me try to clarify my previous comment. If $\phi$ is assumed from the outset to be supported on a compact set $K$ (which is easily arranged), then the relevant equation begins $\int_C f_\epsilon(z)\,dz = \int_C \int_{\epsilon K} f(z-w)\phi_\epsilon(w)\,dw\,dz$. Now there is no possible complaint about changing the order of integration here because we have a jointly continuous function over the compact product $C\times(\epsilon K)$. | |
| Sep 29, 2013 at 16:08 | comment | added | booksee | Sorry, my question was not well-posed. The correct question is: since $z-w$ would go over circles everywhere in $\mathbb{C}$ as $w$ go over $\mathbb{C}$, so we must require the function be integrable over $\mathbb{C}\times\mathbb{C}$ instead of $\mathbb{C}\times C$. How do we prove this ? It's easy to prove it is locally integrable in $\mathbb{C}\times\mathbb{C}$. | |
| Sep 29, 2013 at 8:38 | comment | added | Sean Eberhard | Just take $\phi$ (the approximation to the identity) to be compactly supported, and then it's obvious, as $\mathbb{C}\times C$ can be replaced in that equation throughout by $\text{supp}\,\phi\times C$. | |
| Sep 28, 2013 at 16:42 | comment | added | booksee | Surely, I know. Is that function Lebesgue integrable over $\mathbb{C}\times C$ ? | |
| Sep 28, 2013 at 16:37 | comment | added | Sean Eberhard | Fubini's theorem (en.wikipedia.org/wiki/Fubini's_theorem) states that interchanging the order of integration is nice and friendly. | |
| Sep 28, 2013 at 16:08 | comment | added | booksee | Thanks. The proof seems smooth, but how do you prove you can interchange the order of integration in the 4th formula of that proof ? | |
| Sep 27, 2013 at 12:58 | history | answered | Sean Eberhard | CC BY-SA 3.0 |