Minimizing and maximizing the joint space-spatial frequency uncertainty of Gabor-like functions: comment

Area of Science:

• Signal Processing
• Mathematical Physics
• Quantum Mechanics

Background:

• Joint uncertainty relations quantify the fundamental limits of simultaneously localizing a function in both space and spatial frequency domains.
• Hermite functions, derived from Gaussian functions, are often considered in signal processing and quantum mechanics due to their unique properties.

Purpose of the Study:

• To critically evaluate the assertion that Hermite functions minimize joint uncertainty in space and spatial frequency.
• To investigate the uncertainty-minimizing properties of Hermite functions within a broader class of functions.

Main Methods:

• Mathematical analysis of Hermite functions and their relationship to Gaussian functions.
• Derivation and analysis of the joint uncertainty relation for functions of the form: polynomial multiplied by a Gaussian.
• Comparison of the uncertainty-product values for Hermite functions against other functions in the specified class.

Main Results:

• The study disputes the claim that Hermite functions minimize joint uncertainty.
• Hermite functions were found to maximize, not minimize, the uncertainty product for functions comprising an mth-order polynomial multiplied by a Gaussian.
• This finding contrasts with the conventional understanding and application of Hermite functions in uncertainty principle contexts.

Conclusions:

• Hermite functions do not represent the optimal solution for minimizing joint uncertainty in space and spatial frequency for the analyzed function class.
• The results necessitate a re-evaluation of the role of Hermite functions in contexts where uncertainty minimization is presumed.
• Future research should explore alternative functions or conditions that may achieve true uncertainty minimization.