An uncertainty principle for finite frames

We consider the notion of uncertainty for finite frames. Using a difference operator inspired by the Gauss–Hermite differential equation we obtain a time-frequency measure for finite frames. We then find the minimizer of the measure over all equal norm Parseval frames, dependent on the dimension of the space and the number of elements in the frame. Next we show that given a frame one can find the dual frame that minimizes this time-frequency measure, generalizing some work of Daubechies, Landau and Landau to the finite case and extending some recent work on Sobolev duals for finite frames.