Timeline for bounds on derivatives of mollifiers/mollified functions
Current License: CC BY-SA 4.0
9 events
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| Oct 12, 2019 at 12:25 | comment | added | user58955 | @MateuszKwaśnicki Oh I see, thanks! | |
| Oct 12, 2019 at 8:30 | comment | added | Mateusz Kwaśnicki | (1) You are right, sorry! Still, this leads to an upper bound for the $L^1$ norm of $\phi^{(n)}$. (2) Not sure if I understand correctly; we have $\phi^{(n)}(x) = (-1)^n \pi^{1/4} 2^{n/2} (n!)^{1/2}e^{-x^2/2} \phi_n(x)$, where $\phi_n$ is the Hermite function defined right after (1.2) in that paper. | |
| Oct 12, 2019 at 0:59 | comment | added | user58955 | @MateuszKwaśnicki Thanks for the reference. Two questions: (1) It seems to me that $\phi^{(n)}=(2π)^{-1/2}\exp(−x^2/2)\cdot 2^{-n/2}H_n(x/\sqrt{2})$. Then $\|\phi^{(n)}\|_1$ will depend on $\int |H_n(x)\exp(-x^2)|$ with a different weight function. I think the order of growth should be similar, but is there a straightforward way to derive the result for the weight function $\exp(-x^2)$ from the weight function $\exp(-x^2/2)$? (2) Why is the omitted factor (n!)^{1/2} in the bound from the paper you cited? | |
| Oct 11, 2019 at 20:21 | comment | added | Pietro Majer | For the construction of mollifiers with optimal bounds on derivatives, a very nice reference is Hörmander's The Analysis of Linear Partial Differential Operators, vol I, ch. 1 | |
| Oct 11, 2019 at 19:13 | comment | added | Mateusz Kwaśnicki | Q2: For the standard Gaussian mollifier, $\|\phi^{(n)}\|_1 \sim c 2^{n/2} n^{1/4} (n!)^{1/2}$, see this answer. Furthermore, Lemma 2.1 in this paper states that $\|\phi^{(n)}\|_\infty \le n^{-1/12} (n!)^{1/2}$, giving a reference to Lemma 4.5.2 in [S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes, vol. 42, Princeton University Press, Princeton, 1993]. | |
| Oct 11, 2019 at 15:42 | history | edited | user58955 | CC BY-SA 4.0 |
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| Oct 11, 2019 at 6:57 | comment | added | Mateusz Kwaśnicki | Regarding Q4: $f'' = 2 \delta_0$, so $f * \phi^{(k)} = 2\phi^{(k-2)}$. | |
| Oct 11, 2019 at 6:27 | history | edited | user58955 | CC BY-SA 4.0 |
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| Oct 11, 2019 at 6:21 | history | asked | user58955 | CC BY-SA 4.0 |