Timeline for methods for interpolating a function, holomorphic in the upper halfplane
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
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| Oct 19, 2010 at 13:03 | comment | added | Fiktor | I analyzed conditions on my initial functions and noted two mistakes in the conditions, I have previously written, and, therefore, corrected them. Also I've removed unimportant conditions. About $k(\infty)$: I think, it is ok. It should be equal to 0. | |
| Oct 19, 2010 at 12:58 | vote | accept | Fiktor | ||
| Oct 19, 2010 at 12:58 | history | bounty ended | Fiktor | ||
| Oct 19, 2010 at 12:42 | comment | added | Greg Kuperberg | This is a bit strange. The constraints were changed so that $k(\infty)$ no longer has to agree with $k(-\infty)$. | |
| Oct 19, 2010 at 12:37 | history | edited | Fiktor | CC BY-SA 2.5 |
Correcting inequalities and signs
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| Oct 18, 2010 at 6:38 | history | edited | Fiktor | CC BY-SA 2.5 |
Mistake correction
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| Oct 17, 2010 at 15:10 | comment | added | Fiktor | Function comes from experiment and "it's not likely that function will exceed 30". Probably this doesn't help, so I can exclude this statement. | |
| Oct 17, 2010 at 13:21 | comment | added | Dylan Thurston | Is the upper bound $M_1$ for $n(x)$ known, or just that it exists? The latter follows from continuity and the fact it's approaching $1$ at $\pm \infty$. | |
| Oct 16, 2010 at 19:50 | answer | added | Greg Kuperberg | timeline score: 3 | |
| Oct 16, 2010 at 7:38 | history | bounty started | Fiktor | ||
| Oct 15, 2010 at 13:51 | history | edited | Fiktor | CC BY-SA 2.5 |
Style correction
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| Oct 14, 2010 at 22:29 | history | edited | Fiktor | CC BY-SA 2.5 |
Correct format
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| Oct 14, 2010 at 13:38 | history | edited | Fiktor | CC BY-SA 2.5 |
Adding an example
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| Oct 14, 2010 at 12:13 | comment | added | Fiktor | Yes, they are independent, but both can be calculated given only the real part of $f(x)=\sum c_n e^{inx}$ (if $c_n\to 0$ fast enough): $c_n=\lim_{N\to\infty} \int_{-N}^{N} \Re(f(x))e^{inx}/N dx$. I understand, that function cannot be determined by its values in the finite number of points. Nevertheless we are measuring, for example, temperature and then we are interpolating it between the points. Ok, we use some assumptions on the dependence of temperature on time. But it works. | |
| Oct 14, 2010 at 10:16 | comment | added | Piero D'Ancona | Well yes, if the function is in a Hardy space then the two components are Hilbert transforms of each other. What I mean is that this might not be helpful for your problem. Think of this related example: any periodic function $\sum c_ne^{inx}$ with $c_n=0$ for $n\le 0$ and $c_n$ growing at most polynomially (so it can be a distribution) is the trace of the holomorphic function $\sum c_ne^{inx-ny}$. But the two sequences of coefficients $\Re c_n$ and $\Im c_n$ are completely independent of each other | |
| Oct 14, 2010 at 9:44 | comment | added | Fiktor | It is defined on the whole half-plane $\Im z\geq 0$. And yes, I know only something about the trace of the function. But from holomorphy I know some relations, for example this: en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relation | |
| Oct 14, 2010 at 8:39 | history | edited | Fiktor | CC BY-SA 2.5 |
Improving the title
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| Oct 14, 2010 at 8:12 | comment | added | Piero D'Ancona | Maybe the question is not clear: your unknown function is defined on (a part of) the upper complex plane $\Im z \ge 0$, and you know only a discrete set of values of its trace at $\Im z=0$? If so, I think holomorphy plays no role | |
| Oct 14, 2010 at 7:14 | history | asked | Fiktor | CC BY-SA 2.5 |