so I was working on something and bumped into the following question:
Given some $a>0$, does there exist a complete orthonormal system $ (f_n)_{n \in \mathbb{N}} $ of $L^2(0,\infty)$ such that $\sum_{n=1}^\infty \lVert f_n e^{-a \cdot}\rVert_{L^1(0,\infty)}$?
I have no idea whether this is true or not but if it was true that would really help me with what I am working on. So I have been trying to construct an example sequence but no luck so far. The easiest to verify would probably be if I find a sequence such that it works for $a=0$, i.e., an orthonormal basis with summable $L^1$-norm, but of course that is stronger then what I need here.
My go to, due to general lack of creativity, would be to use some standard examples of orthonormal basis from the literature. That would also save the effort of proving the system to be complete. However, it is surprisingly difficult to find pre-calculated $L^1$-norms of standard basis in the literature and I have honestly no idea how to calculate the $L^1$-norm of e.g. Laguerre polynomials. I have done the calculations for the Fourier basis on $(0,2\pi)$, that is easy enough but does not work (the $L^1(0,2\pi)$-nomr there is actually constant in $n$).
If I had an example for a different domain then $(0,\infty)$ and $a=0$ that would also be good, I think I should be able to make that work for $(0,\infty)$ and at least some $a$ via smart transformations. An example of what I mean: Any orthonormal basis of $L^2(0,1)$ corresponds to an orthonormal basis of $L^2(0,\infty)$ via the transformation $f_n(1-e^{-t}) e^{-t/2}$ and one can bound the $e^{-at}$-weighted $L^1(0,\infty)$ norm of the transformed things by the $L^1(0,1)$ norm if $a>1/2$. Something like that should work for other domains too I think.
So does anybody have promising ideas/intuition for constructing an example sequence or a convincing argument why this will just not work? At the moment I am not convinced one way or the other to be honest, I don't see a good reason for it not to work, but then again I don't have any reason to believe that it works (maybe wishful thinking but that doesn't count).