A standard orthonormal basis for L2 (0,1) is given by the Fourier expansion, as described here, for example (Orthonormal Basis of $L^2$). On the other hand, it seems a standard result that the Laplacians eigenfunctions form an orthonormal basis, too.
Since I needed that basis for speeding-up a numerical simulation, I computed them directly: $\phi_j (x) = \sqrt{2} \sin( j \pi x )$ for positive natural $j$, eigenvalues $-(\pi j)^2$.
On the other hand, the standard basis previously linked works very well in my script, while $\phi$ causes some troubles. So, before digging into many lines of code, let's start from a simple point. I am honestly not to sure that $\phi$ truly consituted a basis, maybe I just misunderstood this informal claim I remember from some past course. So I ask:
Does the family $\phi$ constitutes a basis for L2(0,1)?
Assuming you are restricting yourself to functions which vanish at the endpoints: Yes. Indeed, in this case, the Laplacian eigenfunctions coincide with the fourier basis (the cosines are excluded because they do not satisfy the boundary conditions). But the same fact holds in general.