Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful **practical criteria** to determine whether the sup-norm of the basis functions has a uniform bound? I am sure people investigated this but I do not manage to find a useful result. Can someone offer me references, where I can read about this or a connected problem?

I would like to investigate the properties of some fourth order differential operators (with the corresponding boundary conditions) in $C[0,1]$. Since these operators are nice selfadjoint operators with compact resolvent in $L^2[0,1]$, and the corresponding eigenfunctions are analytic, it would be straightforward to ask whether the properties I need can be carried over using the basis.

Edit on 02/18/2011: Though completely unrelated to my question, googling brought me the following interesting paper:

This gives me hope, however, that my question might have interested somebody...