I am reading a paper regarding the pricing of certain financial derivatives making use of the finite element method. In this paper the following weighted Sobolev spaces are introduced:
$W_{0}$ = $ \{ \phi : \int_{\Omega} \phi^2 dx + \int_{\Omega} x^2 (\phi')^2 dx < \infty \} $
$W_{2,0}$ = $ \{ \phi : \int_{\Omega} \phi^2 dx + \int_{\Omega} x^2 (\phi')^2 \int_{\Omega} x^2 (\phi'')^2 dx < \infty \} $
With associated norms $\Vert . \Vert_{0} $ and $\Vert . \Vert_{2,0} $ , $\Omega = (0,a) $ and where the derivatives are take in the weak sense.
The author then proceeds to give the following result:
$\Vert \phi - \pi_{h}\phi \Vert_{0} \leq ch \Vert \phi \Vert_{2,0} $ $\forall \phi \in W_{2,0}$, where $\pi_{h}$ is the piecewise linear interpolant of $\phi$ over [0,a].
My problem is the definition of $\pi_{h}\phi$ as the functions in the wighted space need not even be defined at 0. So far I have managed to show that the functions in $W_{2,0}$ can be approximated by bounded $C^{\infty}$ functions, are bounded and have a classically differentiable version.
I have seen two possible options:
1) Show the embedding $W_{2,0} \hookrightarrow C(\bar{\Omega})$.
2) Show that $\pi_{h}$ is a bounded linear operator and can thus be extended from the bounded $C^{\infty}$ functions to $W_{2,0}$.
However I can't seem to get either of these two methods to work. Any ideas would be appreciated.
