Periodic Splines and their Applications to Approximation Theory

Synopsis: Obtain innovative results and methods for the approximation of periodic functions with periodic splines.

Level: Master/Bachelor.

Description: A spline is a piecewise-smooth function used in approximation theory, known to be an excellent tool for bridging the discrete and continuous setting. As such, it is therefore at the core of many signal processing applications. Spline theory has been developed for decades and is still very active. Splines have typically been recently recognized as the key elements for recent advances in signal reconstructions based on sparsity promoting optimization methods.

In this project, we focus on the approximation and reconstruction of periodic functions, for which periodic splines play a similar role than in the non-periodic setting. It appears that the theory of periodic splines is far from being fully established. This project is the occasion to revisit and learn about the theory of splines, and to obtain innovative results and methods for the approximation of periodic functions. The main goal of this project consists in obtaining new closed-form expressions for elementary Green functions at the foundations of the theory. In cases where closed-form expressions cannot be obtained, computationally efficient evaluation schemes will also be explored.
The project has strong potential for applications in acoustics, and will typically be at the intersection between mathematics and signal processing.

Deliverables:Theoretical results pertaining to the theory of periodic splines, Python scripts for efficient evaluation of periodic splines.

Prerequisites: Analysis I-IV, good calculus skills and a taste for rigorous mathematics. Some knowledge of functional analysis, approximation theory and Python 3 is a plus.

Type of Work: ~80%-90% theory, ~20%-10% programming depending on the student.