### Finite Frames and Frame Potentials

Directed by Kasso Okoudjou

In this project, students will work to design and analyze systems for effective and efficient signal and data representations. The problems students will investigate are related to frame theory. A finite frame of a d−dimensional vector space ℍ is a spanning set, which allows the analysis and synthesis of vectors in a way similar to basis decompositions. However, frames are redundant systems and as such the reconstruction formulas they provide are not unique. This redundancy plays a key role in many applications of frames which appear now in a range of areas that include, but is not limited to, signal processing, quantum computing, coding theory, and sparse representations, see [2, 4, 5, 6] for an overview.

One set of problems concerns the extremal sets of vectors for the family of p^{th} potential functionals
defined as follows. Given N unit-norm vectors {f_{k}}_{k=1}^{N} in a d−dimensional vector space ℍ, and given
p ∈ (0,∞], the p^{th} frame potential is

When p = 2, FP_{2,N,d} is called the frame potential and its minimizers are called finite unit norm tight
frames (FUNTFs) [1]. Examples of FUNTs include the platonic solids.

The characterization of the minimizers of the p^{th} frame potentials in the case ℍ = ℝ^{2} is still incomplete
([3]), and will be one of the problems students will investigate. More generally, the students
will be asked to characterize the minimizers of FP_{p,N,d} when d ≥ 3. In particular, they will
search for universal minimizers, that is, sets that are optimal for a large range of p when N
and d are fixed? Students will attempt to construct and/or characterize the minimizers of
the p^{th} frame potentials for specific values of p, e.g., the case p = ∞ is very interesting and
subject to a number of fascinating open questions. Students will explore these questions both
theoretically and numerically. Students interested in this project are expected to have some
familiarity with linear algebra, multivariable calculus, and some computer programing, e.g.,
Matlab.

This project will be directed by Dr. Kasso Okoudjou, who is passionate about introducing undergraduate students to research in the mathematical sciences. He is excited by the opportunity to share his love of mathematics with you this summer.

### References

[1] J. J. Benedetto and M. Fickus. Finite normalized tight frames. Adv. Comp. Math., 18(2-4):357–385, 2003.

[2] P. G. Casazza and G. Kutyniok. Finite Frames: Theory and Applications. Applied and Numerical Harmonic Analysis. Springer-Birkhäuser, New York, 2013.

[3] X. Chen, E. Goodman, V., Gonzalez, R. Zhang, and K. A. Okoudjou. Optimal configurations for a family
of p^{th} frame potentials. Adv. Comput. Math, 46(4), 2020. Link to arXiv file of this article.

[4] J. Kovacevic and A. Chebira. Life beyond bases: The advent of frames (part i). Signal Processing Magazine, IEEE, 24(4):86–104, 2007.

[5] J. Kovacevic and A. Chebira. Life beyond bases: The advent of frames (part ii). Signal Processing Magazine, IEEE, 24(5):115–125, 2007.

[6] K. A. Okoudjou, editor. Finite Frame Theory: A Complete Introduction to Overcompleteness, volume 73 of Proceedings of Symposia in Applied Mathematics. AMS, Providence, RI, 2016.