Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question.
Let the periodic function$$\alpha(x+1)=\alpha(x),\quad\alpha(x)>0,\quad x\in{\bf R}$$ and the sequence $$\alpha_n=\alpha(nx)\quad n\in{\bf N}$$ Consider the Hilbert space $$H^1_0([0,1]):=\{u:[0,1]\to{\bf R}\,|\,u,u'\in L^2([0,1]), u(0)=u(1)=0\}.$$
Here is my question:
What kind of convergence can one expect for the sequence $(\alpha_n(x))_{n=1}^{\infty}$, and what is the corresponding limit?
Edit: According to Qiaochu's comment, I assume TWO different inner products here: $$\langle u,v\rangle_1=\int_{0}^1uvdx$$ and $$\langle u,v\rangle_2=\int_{0}^1uvdx+\int_{0}^1u'v'dx$$
For what topology can one expect the convergence of the above sequence?
Edit: If one defines $$\hat{\alpha} = \frac{1}{\int_0^1\frac{1}{\alpha(x)}dx}$$ can one expect some relationship between $(\alpha_n)$ and $\hat{\alpha}$?