I have a process where $z$ varies with $x$. $z$ is well described by a fractal function given by

$z(x)= K \Big( \frac{H}{K} \Big)^{D-1} \sum_{n=0}^{\infty} cos\Big(\frac{2 \pi \beta^n x}{K}\Big) \frac{1}{\beta^{(2-D)n}}$

$K>0$ is a constant which I know the value of.


$ 1<D<2$,

$1< \beta < 2$

The power spectral function of $z$ is

$\hat{P}(\omega)= \frac{K^2}{2} \Big(\frac{H}{K}\Big)^{2(D-1)} \sum_{n=0}^{\infty} \delta(\omega -\frac{\beta^n}{K}) \frac{1}{\beta^{2n(2-D)}}$

Where $\delta(x)$ is the dirac delta function.

I have collected data on the value of $z$ over a range of $x$. $z$ is measured for 500 values of $x$ which have a spacing of $\Delta x=0.0002$ between them. Since it is a fractal structure there are changes at scales smaller than 0.0002 which I cannot measure but I should be able to infer them from what is happening at larger scales.

How can I use the data I have collected to find the values of $K$, $H$, and $\beta$ that best fit it?


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