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.

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