On operators providing shift-invariant uncertainty limit

The uncertainty relation is introduced by using operators which are not necessarily self-adjoint. In general, the uncertainty (lower) limit is different for different functions. Operators are found that define the second moments and the uncertainty limit for their product such that the limit is the same for functions obtained from each other by applying unitary support-size preserving operators, e.g. by shifting in time and frequency. In this case, for all functions of finite energy the limit is related to the minimum support size in the time-frequency domain. Moreover, it is possible to find the discrete-time counterpart of the classical uncertainty relation for continuous-time functions. The operator calculus presents results from signal analysis in a more standard form extensively used in quantum mechanics.