I am not familiar with your particular suggestion, so some general remarks concerning the usefulness of transformations.
Typically, you would like the transformation to preserve some property of the function, or maybe transform some calculation into a different type of calculation, and it should have nice properties.
Fourier transform, for example:
1. is invertible
2. is linear
3. does not change the function's $L^2$-norm
4. turns convolution into product
You would want to establish more such properties; invertibility (or characterization of the kernel) in a suitable function space, linearity, continuity in some suitable norms, what kind of structure it preserves, what kind of operations it interacts with.
This would be building the theory for the transform.
Ideally, you would also find some problem it solves, better or differently than existing tools.
Many transforms have a motivation or interpretation in physics or geometry, too. This suggests applications and intuition for how the transform will work, and encourages investigating it, rather than any arbitrary transform one could define.
tommi
·
2026-05-08T11:29:35Z (4 months ago)