How can I find the smoothest complex analytic function $$x+yi \to u+vi$$ with a finite set of points having prescribed values $$\{x_1,\cdots,x_n\},\{y_1,\cdots,y_n\},\{u_1,\cdots,u_n\},\{v_1,\cdots,v_n\}$$
This question is mostly regarding algebraic or analytical methods to find a solution.
Own work
Being a MSc EE specialized in applied mathematics, I have with linear algebra managed to find approximants by expressing the Cauchy-Riemann equations $$\frac{du}{dx} = \frac{dv}{dy}$$ $$\frac{du}{dy} = -\frac{dv}{dx}$$ on a discretized Cartesian grid with added smoothness regularizing term using the vector-gradient $+\epsilon_1\|\nabla v\|$ as well as a small Tikhonov regularization term $\epsilon_2\|v\|$
For example here we force $0.5 \to 2$. The image represents complex numbers in a way so that the non-black color purple corresponds to the vector (2,0).
Found solution :
By ocular inspection I suppose this solution is suspiciously close to the affine transformation that rotates around some point somewhere in the neighborhood of $(0.5,0.25)$. Perhaps this would be a suitable exercise for a curious student to prove that a first order polynomial possibly with some extra restrictions has a homomorphism (I have hardly studied abstract algebra.. am I using this word right?) to affine transformations on the plane.