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AIP conference proceedings

Nazarov’s uncertainty principle for linear canonical wavelet transform

Mawardi Bahri

Published Jan 1, 2024• 3 citations
Publisher

Abstract

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.

DOI
10.1063/5.0191778

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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.
The aim of this paper is to derive the uncertainty principle which has implications in signal analysis and in quantum mechanics. First, we derive the definition of the wavelet transform in [Formula: see text]-calculus by using some weight function. Certain properties like linearity, scaling, translation, etc. were discussed. Later on, to illustrate this integral transform several results were derived for the effectiveness and performance of the proposed method. Also, this paper surveys recent applications to establish generalized uncertainty principle.
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 and 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 and The reason why the three lower bounds are only associated with LCT parameters and and independent of and is presented. The results show it is possible that the bounds tend to zeros.