Timeline for Taylor-like expansion for a holomorphic function in non-simply-connected domain
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 14, 2019 at 19:47 | vote | accept | Peter Kravchuk | ||
| May 13, 2019 at 4:41 | comment | added | fedja | @PeterKravchuk J.E.Pascoe suggested an alternative description of $h_n$ in comments that you can find more constructive. The reproducing kernel is still hard to find unless you know some particular convenient orthogonal basis (like powers of $z$ for an annulus centered at the origin). If you have some very particular domain in mind, we can try to optimize a bit. Also, if you have some a priori growth restriction for $f$, it may help to choose the weight and even to get bounds for the error. I was primarily concerned with theoretical possibility. The practical implementation is more delicate. | |
| May 13, 2019 at 4:28 | comment | added | Peter Kravchuk | ... 2) I think in the example that I gave for simply-connected $\Omega$ the set of basis functions is essentially invariant under holomorphic maps, i.e. if we map $\Omega$ to $\Omega'$ and $p$ to $p'$ then the functions $h_m$ in two cases differ at most by phase. In your construction there is an explicit dependence on the measure. I am not sure how to formalize it, but do you think there is a natural choice of the measure, given $\Omega$? | |
| May 13, 2019 at 4:27 | comment | added | Peter Kravchuk | Thanks a lot! I have a couple of questions: 1) how would one go about constructing $h_m$ for a given $\Omega$? is it feasible at least in simple examples? maybe there is some natural choice of the weight under which multiplication by $z$ has a simple adjoint, i.e. a finite-order differential operator? ... | |
| May 13, 2019 at 1:52 | history | edited | fedja | CC BY-SA 4.0 |
added 1 character in body
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| May 13, 2019 at 1:06 | history | answered | fedja | CC BY-SA 4.0 |