Timeline for Approximating a smooth function under some restrictions
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 14, 2022 at 21:58 | vote | accept | masala | ||
| Jul 14, 2022 at 20:28 | answer | added | Willie Wong | timeline score: 1 | |
| Jul 14, 2022 at 18:40 | comment | added | masala | @WillieWong I wanted to have the approximation sequence to be in the same Holder ball, to be aligned with the use in the papers I read. But allowing for $p_n$ to be in a bigger Holder ball $C^{m,\alpha}_{M'}$ for some $M'>M$ should be enough to justify the strategy used in the existing literature. | |
| Jul 14, 2022 at 18:36 | comment | added | masala | @WillieWong Yes the unspecified norm was the uniform norm. I have added it to the Question. Sorry for the confusion. | |
| Jul 14, 2022 at 18:34 | comment | added | masala | @IosifPinelis Thank you for pointing these out! $\alpha$ is now included in the definition of $\|g\|_{C^{m,\alpha}}$. The condition is also relaxed to $p_n \in C^{m,\alpha}_{M'}$ for $M'$. | |
| Jul 14, 2022 at 18:31 | history | edited | masala | CC BY-SA 4.0 |
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| Jul 14, 2022 at 18:03 | comment | added | Willie Wong | Reading @IosifPinelis's comment, I suppose another possibility is that the unspecified norm in your Question is the uniform norm, and you are interested in whether you can require the approximation sequence be drawn from the same Holder ball. Is that the correct interpretation? | |
| Jul 14, 2022 at 17:57 | comment | added | Willie Wong | Do you mean trying to approximate in Holder norm? The answer is no. Even $C^\infty$ is not dense in $C^{k,\alpha}$. See e.g. mathoverflow.net/questions/29869/… But you can get approximation in slightly worse space (replace $\alpha$ by $\beta\in (0,\alpha)$. (At least by smooth functions, I think (but may be wrong) should also work for polynomials. See mathoverflow.net/questions/244377/… ) | |
| Jul 14, 2022 at 17:57 | comment | added | Iosif Pinelis | This is still two questions. Also, $\alpha$ is missing in the definition of $\|g\|_{C^{m,\alpha}}$. Also, you probably want to relax the condition $p_n \in C^{m,\alpha}_M([0,1])$ to something like $p_n \in C^{m,\alpha}_{2M}([0,1])$. | |
| Jul 14, 2022 at 17:28 | comment | added | masala | @IosifPinelis Sure! I have revised to keep the most essential question. | |
| Jul 14, 2022 at 17:24 | history | edited | masala | CC BY-SA 4.0 |
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| Jul 14, 2022 at 17:13 | comment | added | Iosif Pinelis | Asking multiple questions in one post is not encouraged on MathOverflow. Also, this strategy is not beneficial for askers, in most cases. | |
| Jul 14, 2022 at 16:31 | history | asked | masala | CC BY-SA 4.0 |