The initial title was asking for connection with Uncertainty Principle, which I changed later as per S. Carnahan's suggestion. I also added what led me to think that there might be connection to results in Harmonic Analysis :-
That Polya's result about entire function of exponential type, having slow growth (i.e., type < log 2 and taking integer values at integers have to be polynomial can be seen as a Uncertainty type result.
If someone could elaborate on what I am missing I would be very thankful.
I left a comment with my evaluation of the question, and voted to close. The question was later considerably improved, and I deleted my critical comment as no longer applicable, but forgot to undo my vote to close (actually, I'm not entirely sure it's possible to retract a vote). Perhaps it should be reopened.
Thank you Prof. Myerson. I noticed you took the trouble to proofread and edit my question too. As a non native speaker of the language I end up making some obvious grammatical mistakes. It really helps me to improve when someone points out these mistakes.