New Inequalities and Uncertainty Relations on Linear Canonical Transform Revisit

The uncertainty principle plays an important role in mathematics, physics, signal processing, and so on. Firstly, based on definition of the linear canonical transform (LCT) and the traditional Pitt’s inequality, one novel Pitt’s inequality in the LCT domains is obtained, which is connected with the LCT parameters a and b. Then one novel logarithmic uncertainty principle is derived from this novel Pitt’s inequality in the LCT domains, which is associated with parameters of the two LCTs. Secondly, from the relation between the original function and LCT, one entropic uncertainty principle and one Heisenberg’s uncertainty principle in the LCT domains are derived, which are associated with the LCT parameters a and b. The reason why the three lower bounds are only associated with LCT parameters a and b and independent of c and d is presented. The results show it is possible that the bounds tend to zeros.