methods for interpolating a function, holomorphic in the upper halfplane

Question

Let $n,k\colon\mathbb{R}\to\mathbb{R}$ be real functions such that function $N$ given by $N(x)=n(x)-ik(x)$ is a holomorphic function in the upper half-plane. Also I know some additional properties of these functions:
1. $n(x)=n(-x)$ if $x\in\mathbb{R}$;
2. $k(x)=-k(-x)$ if $x\in\mathbb{R}$;
3. $k(x)>0$ if $x>0$;
4. limit $\lim_{x\to\infty}(n(x)-ik(x))$ exist.

Unfortunately I don't know functions $n$ and $k$. I know only some approximate values of $n_i\simeq n(x_i)$, $k_j\simeq k(y_j)$ for finite number of points $x_i,y_j\in [a,b]\subset\mathbb{R}$. Which methods can I use in order to get approximate values of the function $N$ on the whole region $[a,b]$? Where can I read about them?

Update (example data and an experiment):

 x_j      n(x_j)
 0.000  1.36364  
10.204  1.32231  
17.346  1.23967  
18.367  1.19835  
27.551  0.53719  
31.632  0.53719  
37.755  0.82644  
42.857  0.90909  
47.959  0.95041  
50.000  1.32231  
51.020  1.36364  
53.061  2.14876  
56.122  2.60331  
61.224  2.47934  
64.285  2.06612  
67.346  1.40496  
70.408  1.36364  
83.673  1.94215  
90.816  1.98347  
102.04  1.94215  

y_j     k(y_j)
0.000   0.00095  
3.061   0.00288  
8.163   0.00425  
11.22   0.00870  
14.28   0.01562  
24.48   0.09638  
27.55   0.18459  
30.61   0.40261  
32.65   0.52213  
47.95   1.47684  
50.00   1.68181  
53.06   1.47684  
71.42   0.12499  
78.57   0.07931  
102.0   0.04715  

I've tried to fit a $n(x)$ by $\Re(f(x))$ with $f(x)=1+\sum_i\frac{c_i^2}{x-a_i-i b_i^2}$ with $c_i,a_i,b_i\in \mathbb{R}$. Here is the result of the fitting:

Not a good fitting :(. This is actually why I ask this question. But note, I haven't used the data I have for k(x). Let's check, how is it approximated by $\Im(f(x))$ (with coeffitients from the previous fitting).

It's better, than nothing.

Answer 1

If I understand correctly, we can make a change of variables from $z$ in the upper half plane to $\zeta$ in the unit disk. Then the function $N(\zeta)$ is continuous on the unit circle $\zeta = \exp(i \theta)$ and has a unique harmonic extension to the unit disk. If $N$ is a favorable function, then this harmonic extension is complex analytic. A function in this class is one with a Taylor series in $\zeta$ with a radius of convergence at least 1. Or, it is a Fourier series in $\theta$ which uses only terms $\exp(i k \theta)$ with $k \ge 0$, and whose coefficients satisfy a convergence condition.

In your question, in addition, the real and imaginary parts of $N(\zeta)$ satisfy certain inequalities, and they are supposed to be as close as possible to certain data points. Let's assume a least-squares error model for these data points. If you put all of this information together, the result is that the Fourier coefficients $a_k$ lie in a certain convex region, and you want to minimize a certain quadratic objective function. Thus, your question is a convex programming problem. You can make this into a practical computational convex programming problem by assuming that $k \le 100$ (say) and also discretizing the unit circle on which $\zeta$ lies. Or, there are also numerical methods to impose your constraints (1) and (2) without discretizing the unit circle, but they make the convex programming problem more complicated.