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.

uniformbound on the variance. $\endgroup$ – Steve Flammia Mar 12 '10 at 17:10