Some time ago, I came across an article due to Iosevich, Lai and Mayeli titled Tight wavelet frame sets in finite vector spaces. I’m not exactly sure why the suggestion system of ScienceDirect thought I might like this article from Applied and Computational Harmonic Analysis, but I was intrigued. For one, because Ingrid Daubechies, the person commonly accredited with the development of wavelets for applications, is an alumnus of Vrije Universiteit Brussel, where I am currently at. Secondly, finite vector spaces is something I am familiar with, so why not take a look?
It turned out that there was a particular problem in the paper posed as an open question on the very last page:
“Question #1. Does there exist tight frame wavelet sets when ?”
I managed to solve this question, and I will record it here, mainly for my own convenience. The what and why of tight frame wavelet sets is not very important, the most important aspect is the fact that someone from a finite geometrical background can contribute to problems in harmonic analysis. We have more in common than we think!
Going through the paper, it turns out that to solve this problem, all one needs is a multiplicative tiling of , i.e. a set of non-zero vectors and a set of automorphisms such that every non-zero vector in can be uniquely written as . This is a ‘multiplicative’ analogue of translational tilings which consists of two sets of vectors such that every vector in can be uniquely written as .
When the authors took to be a set of representatives of the circles defined by , which are the orbits of under a group of rotations (which can be written down explicitly). The problem with is that the circle with radius zero consists of more points than just the origin, as is a square!
This problem with squares and non-squares appearing in computations over finite fields is very common and working around it often uses a solid geometrical background. In fact, the obstacle here can be circumvented by considering ellipses defined by instead, where is a non-square in . One can do the computations and see that after some adaptions, everything works out just fine. However, there is an even more elegant approach to this construction, which is the one we will show.
Instead of considering , we can consider the isomorphic vector space . We will make the isomorphism explicit by constructing the field extension as (where is the same non-square as before). In this way, if such that , we can identify every element with . In this bigger field, we can consider the standard norm function .
Now take to be a set of points, one from each of the norm curves , and the set of points with norm (which is in fact a subgroup). Then it is immediately checked that is a multiplicative tiling set, and these computations are reduced to one-line proofs, as opposed to the work that is necessary when working in (which takes 2 pages in the aforementioned paper). Finally, adapting this multiplicative tiling into a tight wavelet frame set works exactly the same as in the paper, which completely answers their question.