The Uncertainty Principle: A Survey and Exploration of Orthonormal Functions

Abstract

We give a survey of the classical uncertainty principle and uncertainty principles related to orthonormal sequences of functions. Furthermore, we present a new result concerning the minimality of additive uncertainty for n orthonormal functions, namely that it is achieved by linear combinations of the n first Hermite functions. Next, we conjecture that the minimal multiplicative uncertainty for two orthonormal functions is achieved by a special linear combination of the two first Hermite functions. Lastly, we discuss relations to Gabor superframes and make some observations regarding the stability of multiplicative uncertainty measure.