Suppose we have some 2D points $x_i$ (which we may take to be complex numbers) and some corresponding 2D points $y_i$. We seek a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x_i)\approx y_i$, and $f$ has to be conformal / holomorphic.
(it suffices that $f$ is conformal in the convex hull of the $x_i$, it doesn't have to be conformal everywhere)
Now there is a straightforward way of doing this: just use the Lagrange polynomial. A polynomial can be constructed for which all $f(x_i)=y_i$, and, being polynomial, it is also holomorphic and thus conformal. Problem solved, right?
The disadvantages of the Lagrange polynomial are well known, and lead me to suspect that this is not actually a good solution. Is there a better way of doing this?
One solution would be to use Spline functions. Depending of your data, it can be done in 2D 1. The
scipyfunction proposes a numerical implementation here if the data is know on a cartesian mesh.If you know that your data has a compact support, or more precisely if is as: $$ \lim_{|z| \rightarrow + \infty} \exp(z^2) f(z) = 0,$$ then you could use the Hermite transform. This is quite tractable in 2D 2. This python module is doing it: Bernard
1 Opfer, G., Puri, M.L., 1981. Complex planar splines. Journal of Approximation Theory 31, 383–402. https://doi.org/10.1016/0021-9045(81)90104-0
2 Wünsche, A. (2001) Hermite and Laguerre 2D Polynomials. Journal of Computational and Applied Mathematics, 133, 665-678. http://dx.doi.org/10.1016/s0377-0427(00)00681-6