Uncertainty Principle and Elementary Wavelet

Alexander I. Saichev, Wojbor A. Woyczyński

Wojbor A. Woyczyński

Firdous A. Shah, Azhar Y. Tantary

Jack G. Dodd

Sabrine Arfaoui, Maryam G. Alshehri, Anouar Ben Mabrouk

In the present paper, an uncertainty principle is derived in the quantum wavelet framework. Precisely, a new uncertainty principle for the generalized q-Bessel wavelet transform, based on some q-quantum wavelet, is established. A two-parameters extension of the classical Bessel operator is applied to generate a wavelet function which is used for exploring a wavelet uncertainty principle in the q-calculus framework.

Saifallah Ghobber, Hatem Mejjaoli

The deformed wavelet transform is a new addition to the class of wavelet transforms that heavily rely on a pair of generalized translation and dilation operators governed by the well-known Dunkl transform. In this study, we adapt the symmetrical properties of the Dunkl Laplacian operator to prove a class of quantitative uncertainty principles associated with the deformed wavelet transform, including Heisenberg’s uncertainty principle, the Benedick–Amrein–Berthier uncertainty principle, and the logarithmic uncertainty inequalities. Moreover, using the symmetry between a square integrable function and its Dunkl transform, we establish certain local-type uncertainty principles involving the mean dispersion theorems for the deformed wavelet transform.