Quantitative uncertainty principles associated with the \(k\)-generalized Stockwell transform (Q2148337)

The Stockwell transform provides frequency-dependent resolution. It is a generalization of the Fourier transform and is regarded as a multi-scale local Fourier transform. The authors of the present paper define the \(k\)-generalized Stockwell transform, and then study its several properties such as Parseval's formula, Plancherel's formula, Calderon's reproducing formula and the inversion formula of the \(k\)-generalized Stockwell transform. The authors also derive several Heisenberg-type uncertainty principles (uncertainty inequalities) for the \(k\)-generalized Stockwell transform. Here, each Heisenberg-type uncertainty inequality is a generalization of the Heisenberg uncertainty inequality in quantum mechanics associated to the Fourier transform. Moreover, the authors obtain concentration uncertainty principles and weighted uncertainty principles for the \(k\)-generalized Stockwell transform.