I think this is also a good reference: google.co.kr/books/edition/Spline_Models_for_Observational_D‌​ata/…

Well I try to answer my own questons whenever I can... but I don't think I have enough knowledge to write a modern treatment =D.

Not accessible to me. Also is more of an engineering book (from the title and google preview it seems at least). I was more hoping for a rigorous mathematical treatment.

I'll wait for a more conclusive answer.

What if I assume maybe $f \in W(\mathbb{R}^2)^{3,2} \cap C^3(\mathbb{R}^2)$? (so first 3 order derivatives in $L^2$ but also continuous?) Is it too restrictive?

@Kostya_I I thought of that but I wonder if there's something like Banach algebra style. With that I mean you can derive the fourier transform in $L^1$ by finding the set of all homomorphism $\left\{ \phi \right\}$. Namely you go with $\phi(xy) = \phi(x)\phi(y)$ a bit of manipulations and you end up with the fourier transform. Don't know if this make sense but this is what I was going for. Homomrphism for groups are well defined, I thought maybe there's something there?

Cause I have a time series data of rotations expressed as matrices/quaternions or equivalent. Because of that target space I abstracted the question in the one I've asked. Just wondering if someone has thought about it really.

Hopefully is clearer now.

I guess but because lie groups have a bit of more structure I thought it was more likely to get an answer.

@RobPratt Maybe you can give a full answer or a reference as I don't understand the transformation from non linear to linear (I mean I got a rough idea, but probably more details would be useful as a reference).