Uncertainty Principle and Wavelet Transforms

Wojbor A. Woyczyński

Firdous A. Shah, Azhar Y. Tantary

M. T. Bliznetsov

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.

Mawardi Bahri

The linear canonical wavelet transform is a useful tool for image analysis. It may be considered as an extension of the conventional wavelet transform in the context the linear canonical transform. In the current study, we introduce the definition of the linear canonical wavelet transform and collect its essential properties. Based on the properties we derive in detail Nazarov’s uncertainty principle related to this generalized wavelet transformation.

Hicham Banouh, Anouar Ben Mabrouk

In the present work, we are concerned with the development of a new uncertainty principle based on the wavelet transform in the Clifford analysis/algebra framework. We precisely derive a sharp Heisenberg-type uncertainty principle for the continuous Clifford wavelet transform.