For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define the Sobolev space $L^2_k$ on $M$, we can just define $<f_{i},f_{j}>=0$ for $i\neq j$ and $\langle f_i,f_i \rangle =\lambda_i^k$ and then do completion. In this case, $(f_1,f_2,\ldots)$ are orthogonal in any $L^2_k$ inner product and any function in $L^2_k$ can be approximated by their finite sum.
My question is: For manifold with boundary. Do we have similar constructions? I.e. can we find a set of smooth functions such that (1) They are pependicular under (suitable defined) $L^{2}_{k}$-inner product for all $k\geq 0$. (2) Any function in $L^2_k$ can be approximated (in $L^2_k$ norm) by finite sum of these smooth function.
I am most interested in the easiest case: the manifold is just the closed interval $[0,1]$ so if anyone knows the answer for this special case that would be also great for me.
