An exercise with orthonormal basis

While writing the general case of the large sieve, one question of minor interest arose, which provides a nice exercise (or exam problem) for a course on finite-dimensional Hilbert spaces.

Since it’s not yet possible (as far as I can see, but I will try to investigate the issue) to include either LaTeX formulas (in the style used in a number of WordPress math blogs) or MathML formulas in the ETH blogs, I’ve resorted to the rather embarassing option of using dvipng to produce an image with the LaTeX content of this post…

Coming back to regular HTML, where one can make links, here’s one to the short note I wrote on this, with the proof of the result indicated. Note that I would be surprised is this were really at all new. There’s one lingering question that I haven’t answered at the moment: does there exist a proof by pure thought that, for the uniform density, there is an orthonormal basis of functions with constant modulus 1?