Geometric Deep Learning - G-Smoothing operator on polynomials

(Note: My question resolves about a problem stated in the following lecture video: https://youtu.be/ERL17gbbSwo?t=413

Hi,

I hope this is the right forum for these kind of questions. I'm currently following the lectures of geometric deep learning from (geometricdeeplearning.com) and find the topics fascinating. As I want to really dive in I wanted to also follow up on the questions they state towards the students. In particular my question revolves around creating invariant functions using the G-Smoothing operator (To enforce invariance, for a particular hypothesis class, you smooth over all possible group elements, e.g. rotations for a 2D-grid/Image). They pose a question about the result of the G-Smoothing operator on k-degree polynomials with d dimensional features on a simple circular grid with the symmetry group of being the cyclic group of order d. I'm not quite sure how to approach this problem, because I do not have a strong formal mathematical background. I started thinking first of individual functions of the hypothesis class and observing the result of the smoothing operator. Since every possible coefficients of the polynomials are possible this would result in the set of Reals if I'm not mistaken. But I think they want a more formal approach to the problem. As they do not give any other examples I was wondering if someone here came across this particular problem and could give some advice.

PS: If you think there is another platform better suited for this kind of question please let me know!