Student Mentoring

I have some interesting problems in Harmonic analysis for LSU undergrads that we can work on together. Shoot me an email if you like to discuss any of this.

Uncertainty Principle: The uncertainty principle, originally formulated in quantum mechanics, also has significant analogs in mathematical analysis, particularly in signal processing and Fourier analysis. In the context of analysis, it states that a function and its Fourier transform cannot both be highly localized or sharply confined at the same time. In simple terms this means, function and its Fourier transform cannot be simultaneously ”small”.

This is similar in spirit to Heisenberg's uncertainty principle in quantum mechanics, where position and momentum (akin to time and frequency) cannot both be precisely known simultaneously. There are several versions of the uncertainty principle known to us now. In this context, we can work together some of these projects. Here I have a list of some topics:

### Uncertainty Principle in Fourier Series

Objective: Investigate how the uncertainty principle applies to Fourier series representations of periodic functions.

Description:

Study how the localization of a function on the interval (spatial or temporal localization) affects the localization of its Fourier coefficients in the frequency domain.

Use periodic functions like square waves, sawtooth waves, or trigonometric polynomials, and analyze how their sharpness in the time domain leads to the spread of their Fourier coefficients.

Visualize this relationship and discuss the implications for the uncertainty principle in periodic settings.

Skills: Fourier series, harmonic analysis, Python or MATLAB for visualization.

### Paley-Wiener Theorem and Uncertainty Principle

Objective: Study the Paley-Wiener theorem, which connects the uncertainty principle with analytic functions in harmonic analysis.

Description:

Understand the statement of the Paley-Wiener theorem, which relates the support of a function and its Fourier transform.

Investigate how this theorem imposes a restriction on how sharply a function and its Fourier transform can be localized.

Apply the theorem to specific examples and functions (such as analytic functions) to see how the uncertainty principle manifests.

Explore the consequences of the Paley-Wiener theorem in signal processing and physics.

Skills: Harmonic analysis, complex analysis, Paley-Wiener theorem.