Timeline for Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 7, 2014 at 7:36 | comment | added | Piero D'Ancona | $v=\Delta u_j$ is a functional on $H^1$, which vanishes on all elements of an orthogonal basis with the exception of $u_j$. The space of functionals with this property is one dimensional and must be generated by $u_j$ | |
| Mar 7, 2014 at 6:47 | comment | added | user47808 | The second bracket is a duality bracket, how does it ensure that $\Delta u_j$ is a multiple of $u_j$? | |
| Mar 6, 2014 at 17:26 | comment | added | Michael Renardy | However, if we are content with orthogonality with respect to an equivalent inner product in one of the spaces, then it can be done. For a lot of purposes, this might be enough. | |
| Mar 6, 2014 at 14:47 | comment | added | Piero D'Ancona | $\Delta u_j$ is in $H^{-1}$ since $u_j$ is in $H^1$ by assumption | |
| Mar 6, 2014 at 14:34 | comment | added | Nate Eldredge | How do you know $u_j$ is in the domain of the Laplacian? | |
| Mar 6, 2014 at 10:18 | vote | accept | CommunityBot | ||
| Mar 6, 2014 at 10:18 | vote | accept | CommunityBot | ||
| Mar 6, 2014 at 10:18 | |||||
| Mar 6, 2014 at 9:47 | history | answered | Piero D'Ancona | CC BY-SA 3.0 |