Timeline for Linear interpolation in weighted Sobolev spaces
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 11, 2014 at 14:36 | comment | added | User76765 | Perhaps I am missing something, but the density of $C^{\infty}$ makes use of the weighted Sobolev norm. So, each function $\phi \in W_{2,0}$ may be approximated by a sequence of $C^{\infty}$ functions in the weighted norm. I don't see how this yields uniform convergence? | |
| Sep 11, 2014 at 14:06 | comment | added | User76765 | For the boundedness, the standard Soboloev embedding theorems can be used to show boundedness on $(\epsilon,a)$ for and small $\epsilon$ as the weights are non-zero and bounded here. Hence, in essence all we need to show is that $\phi$ is bounded as it approaches $0$. To this end, for each $\phi$ one can find $C > $ such that if $\phi(x) > C$ then $\phi'(x)> 0$ and $D < 0$ such that if $\phi(x) < D$ then $\phi'(x) < 0$. This will clearly lead to a contradiction if $\phi$ is unbounded. | |
| Sep 9, 2014 at 10:31 | comment | added | Michael Renardy | How did you prove functions in $W_{2,0}$ are bounded? Did you not do this by showing that the $L^\infty$ norm is bounded in terms of the $W_{2,0}$ norm? You also say you know $C^\infty$ functions are dense. So you have a uniformly convergent sequence of $C^\infty$ functions. This implies the limit is continuous. | |
| Sep 9, 2014 at 8:17 | comment | added | User76765 | Okay, so I have shown that the interpolant is not bounded, so the embedding seems to be the only option. | |
| Sep 7, 2014 at 14:24 | comment | added | User76765 | If the embedding is untrue, it requires a bounded oscillating function, from which it follows that the derivative will be unbounded and oscillate between $+- \infty$ as x approaches 0, similarly for the second derivative. | |
| Sep 7, 2014 at 14:08 | comment | added | User76765 | It is basically just the right continuity at 0 that is needed, but I couldn't see any way that this followed from boundedness or the density of $C^{\infty}$. | |
| Sep 7, 2014 at 13:23 | comment | added | Michael Renardy | If you can prove that functions in $W_{2,0}$ are bounded and can be approximated by $C^\infty$ functions, then the same proof is very likely to show that these functions are continuous. | |
| Sep 7, 2014 at 13:07 | review | First posts | |||
| Sep 7, 2014 at 13:09 | |||||
| Sep 7, 2014 at 13:07 | history | asked | User76765 | CC BY-SA 3.0 |