Timeline for Resource on Infinite Systems of Difference Equations
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 28, 2013 at 15:46 | vote | accept | 042 | ||
| Jul 21, 2013 at 12:21 | comment | added | 042 | Thank you for the extensive comment. This seems like a path to follow. However, I shall try to leave this question open for some time, mainly in order to seek some other reference on the general theory you have mentioned. | |
| Jul 20, 2013 at 23:32 | comment | added | Igor Khavkine | @042, if you can find a linear transformation $T$ (which does not destroy too much structure of your equation) and $B = T A T^{-1}$ is self-adjoint, then you could solve the equivalent equation $y_{j+1} = B y_j$, with $y_j = Tx_j$. If $A$ cannot be made self-adjoint in this way, you need the more general spectral theory for operators on Banach spaces. Unfortunately, I don't know of a good modern reference, but some information on that can be found in Ch.XI of Functional Analysis by Riesz and Sz.-Nagy. | |
| Jul 20, 2013 at 23:08 | comment | added | Igor Khavkine | You're right, I didn't think the formula completely through when I wrote it. It's fixed now. But it doesn't really change the rest of the answer. | |
| Jul 20, 2013 at 23:07 | history | edited | Igor Khavkine | CC BY-SA 3.0 |
added 46 characters in body
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| Jul 20, 2013 at 17:01 | comment | added | 042 | I have one more question... By $\exp(A)$ do you denote $e^A$? Because this sounds to me more like a solution to a differential equation than to a difference equation. But I am surely missing something... | |
| Jul 20, 2013 at 16:55 | comment | added | 042 | Thank you for your answer, this sounds good... My equations are exactly of the form $x_{i+1} = Ax_i$, however $A$ need not be self-adjoint. In my basic setting, $A$ is a bounded operator on $\ell^{\infty}$ that can be viewed as an infinite matrix that is $k$-diagonal for some constant $k$. However, there may be a possibility to transform a problem to a more usual space $\ell^2$. | |
| Jul 20, 2013 at 16:19 | history | answered | Igor Khavkine | CC BY-SA 3.0 |