New results on the continuous Weinstein wavelet transform

Area of Science:

• Harmonic Analysis
• Functional Analysis
• Time-Frequency Analysis

Background:

• The Weinstein operator is a key component in various areas of signal processing and quantum mechanics.
• Understanding the properties of transforms associated with such operators is crucial for theoretical advancements.

Purpose of the Study:

• To introduce and analyze localization operators for the continuous wavelet transform associated with the Weinstein operator.
• To investigate the concentration properties and uncertainty principles for this transform.

Main Methods:

• Definition and analysis of localization operators.
• Application of techniques from functional analysis to prove boundedness and compactness.
• Analysis of concentration on sets of finite measure.
• Derivation of uncertainty principles (Benedicks-type, Donoho-Stark, and Heisenberg-type).

Main Results:

• Established the boundedness and compactness of localization operators for the continuous wavelet transform of the Weinstein operator.
• Provided analyses of the concentration of the transform on finite measure sets.
• Proved multiple versions of Heisenberg-type uncertainty principles, alongside Benedicks-type and Donoho-Stark uncertainty principles.

Conclusions:

• The study provides a comprehensive analysis of localization operators and uncertainty principles for the continuous wavelet transform of the Weinstein operator.
• The findings contribute to a deeper understanding of the mathematical properties of this transform and its associated operators.