I have a curve that represents a quantum mechanical probability. I am interested in finding an algorithm (or calc I/II type formula) to compute a weighting function, k(), that will reconstruct this curve using truncated cos(). I don't think this is Fourier problem ...
The reconstruction is to study the how quantum probabilities might be represented by coupled classical rotational motion.
The curve itself, f(x)^2, takes two parameters $k_1 , k_2$ as arbitrary inputs, but all other constants are calculated from them.
$f(x)^2=(A cos(k_1 x) + B sin(k_1 x))^2$ for x<0 $f(x)^2=(C cos(k_2 x) + D sin(k_2 x))^2$ for x>=0
Where:
$B={1 \over \sqrt{ k_2 + k_1 }}$
$A=\sqrt{1-B^2}$
$C=A$
$D={k_1 \over k_2} B$
f(x)^2 is defined over $[x_1:x_2]$
$x_1 = {atan(-{A \over B}) \over k_1}$,
$x_2 = {atan(-{C \over D}) + \pi) \over k_2}$
The way I need to reconstruct $f(x)^2$ is by the following summation of truncated cos functions:
$isubpx(x)=scale \cdot \Sigma_{n=1}^{1000} cos_t(2 k(X_n)(x-X_n)) $
Where:
$X_n=x_1+{\pi \over {4 k_1}} + (x_2-x_1-{\pi \over {4 k_1}} - {\pi \over {4 k_2}}){n \over 1000} $
$cos_t(x) = cos(x)$ for $-{\pi \over 2} < x < {\pi \over 2}$
$cos_t(x) = 0 $ elsewhere.
$scale \approx 0.01822385 \cdot k_1$
Note: If constant $k_1=k_2$, then the k() function becomes constant over x. In general, the k() function will likely start at $k_1$ and end at $k_2$
As an example, if I choose $k_1=2$ and $k_2=1$, the following plot shows the function (in purple), and it shows my first attempted weighting function (in blue), and the sum of the 1000 localized cosine functions in green.
The associated cos() function's width is set by the blue line's height; 1/k() -- so the blue line's height measures the area each cos() will add to the green curve.