Timeline for Higher order estimate under trigonometric basis in non-periodic case?
Current License: CC BY-SA 4.0
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| Apr 7, 2019 at 0:53 | comment | added | Yidong Luo | @ VorKir I also trend to the negative answer. I tried to examine the concrete case when $ r = 1 $, that is, $ y \in H^1 (0,2\pi) $. In comparsion between $ \Vert (I-P_n) y \Vert_{L^2}$ and $ \Vert y' \Vert_{L^2} $, I found the latter represented in Fourier coefficients would induce a term as $ y(2\pi) - y(0) $ which makes estimate hard to move on. This is the discontinnuity you specify. But i still feel suspicious if there exists some trick to move on or with a new condition, like evaluation at one endpoint can make this estimate move on? | |
| Apr 6, 2019 at 23:43 | comment | added | VorKir | I'd expect the answer is no though I can't give a precise reference for that. Fourier series for a non periodic function should be the same as Fourier series for its periodic extension which will be discontinuous and thus will have no good rate of convergence. | |
| S Apr 6, 2019 at 22:03 | history | suggested | CommunityBot | CC BY-SA 4.0 |
better MathJax usage, including correct use of \text{}
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| Apr 6, 2019 at 21:27 | review | Suggested edits | |||
| S Apr 6, 2019 at 22:03 | |||||
| Apr 6, 2019 at 11:22 | history | asked | Yidong Luo | CC BY-SA 4.0 |