An uncertainty principle for finite frames
In this article the authors define an uncertainty principle for finite frames. They begin by developing a time-frequency localization measure for the entire frame using a difference operator. Then they prove that a subset of columns of the discrete Fourier transform matrix are optimal for minimizing this measure over all equal norm Parseval frames. This gives a sharp constant for the lower bound which is dependent on the dimension of the space and the number of elements in the frame. The authors also find a minimizer and bounds for the time-frequency measure in the case of Parseval frames which are not necessarily equal norm. Next they show that given any matrix whose columns form a frame, it is possible to generate a dual frame that minimizes the finite time-frequency measure. Finally, to illustrate this, the authors generate a random frame and compare the time-frequency localization of this dual with the canonical one.
scientific article; zbMATH DE number 5804456
scientific article