Simultaneous time-frequency concentration of orthonormal sequences? Does there exist an orthonormal basis of square-integrable functions (either $L^2(\mathbb{R})$ or $L^2(\mathbb{C})$) such that the sequence of functions has bounded variance, and also the sequence consisting of the Fourier transform of each function also has bounded variance?  
Some background:
This question came up in a comment on SciRate regarding a recently translated paper by von Neumann.  There the commenter, Matt Hastings, points out some related results.
In particular, the Balian-Low theorem states that this can't exist for any Gabor basis, i.e. one which is composed of time and frequency translates of a given fiducial $L^2$ function.  If there were a generalization of this theorem to arbitrary bases, it would prove that such a sequence can't exist.  
 A: Such orthonormal bases do exist, as proved in:
Bourgain, J. A remark on the uncertainty principle for Hilbertian basis. J. Funct. Anal. 79 (1988), no. 1, 136--143 (MathSciNet link).
The theorem says that for each $\rho>1/2$ there is an orthonormal basis for $L^2(\mathbb{R})$ such that all of the variances of the basis elements and their Fourier transforms are less than $\rho$.  After the statement Bourgain remarks:

Thus Balian’s strong uncertainty principle does not hold for a nonperiodic
  basis.

It is remarkable that this appears to have been discovered (rediscovered?) well after von Neumann's time.  Powell proved more recently the result that Matt Hastings mentioned, namely that in such a case the sequence of means of the orthonormal basis is unbounded.

My old answer, posted before reading Matt Hastings's comment led me to the correct question, was to the question of whether all of the variances can be finite.  It was this:
Yes, because you can take an orthonormal basis in the Schwartz space by applying Gram-Schmidt to a countable $L^2$ dense subset of the Schwartz space.
A: The Fourier transform on $L^2({\mathbb R})$ has an complete set of eigenvectors (that is, there is an o.n. basis of $L^2({\mathbb R})$ consisting of eigenfunctions for the FT, and they are all in the Schwartz space). Does this do what you want?
