Timeline for Convergence of mollified functions in weighted $L^p$ norm
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 26 at 8:54 | comment | added | Akira | Ah I got it. Thank you for your informative answer. Are you aware of results about convergence rate of this kind? | |
| Jun 26 at 8:51 | comment | added | Mateusz Wasilewski | This is a standard argument using the Uniform Boundedness Principle. We want to show that the sequence of operators $T_n f:= \rho_n\ast f$ converges pointwise to identity. This is equivalent to the fact that the norms of $T_n$'s are uniformly bounded (which is what I am focusing on in this answer) and that the $T_n$'s converge to identity on some dense subset, which is usually the easier part, because you can work with, e.g. smooth functions. | |
| Jun 26 at 7:56 | comment | added | Akira | Our goal is to bound $\int \left| \int \rho_n(x-y) f(y) dy - f(x) \right|^p d\mu(x)$, but your treatment is about $\int \left| \int \rho_n(x-y) f(y) dy \right|^p d\mu(x)$. Could you elaborate more? | |
| Jun 26 at 7:17 | vote | accept | Akira | ||
| Jun 25 at 10:51 | history | answered | Mateusz Wasilewski | CC BY-SA 4.0 |