Let's say I have this function $$f(x,y) = \sum_{i=1}^{n} x_i * y_i$$ and I have the values of $f(A,A)$, $f(A,B)$, $f(A,C)$, f(B,B), ... . Can I somehow get the value of $$f(A,B,C) = \sum_{i=1}^{n} A_i * B_i * C_i$$?
I tried applying "mutated" dot product identity: $$f(A,B,C) = \sqrt{f(A,A)f(B,B)f(C,C)}\cos\theta$$ but couldn't think how to get $\cos\theta$.
NOTE: I think it's related to the dot product above, because when I isolate it, it's value is between $-1$ and $1$.
The actual serieses' values is not present. They all have the same size, and they don't have\share property.
No, you cannot recover the sum $$\sum_{i=1}^\infty x_i y_i z_i$$ from the sums $$\sum_{i=1}^\infty x_i^2,\quad \sum_{i=1}^\infty x_i y_i,\quad \ldots,\quad \sum_{i=1}^\infty z_i^2.$$
To see this, suppose that we have the sequences
$$x: 1, 1, 0, 0, 0, 0, \ldots$$ $$y: 1, 0, 1, 0, 0, 0, \ldots$$ $$z: 0, 1, 1, 0, 0, 0, \ldots$$
then we have $$\sum_{i=1}^\infty x_i^2 = \sum_{i=1}^\infty y_i^2 = \sum_{i=1}^\infty z_i^2 = 2$$ and $$\sum_{i=1}^\infty x_i y_i = \sum_{i=1}^\infty x_i z_i = \sum_{i=1}^\infty y_i z_i = 1$$ as inputs, from which we want the output $$\sum_{i=1}^\infty x_i y_i z_i = 0$$
On the other hand, consider the sequences $$x' : 1, 1, 0, 0, 0, 0, \ldots$$ $$y' : 1, 0, 1, 0, 0, 0, \ldots$$ $$z' : 1, 0, 0, 1, 0, 0, \ldots$$
again we have $$\sum_{i=1}^\infty (x'_i)^2 = \sum_{i=1}^\infty (y'_i)^2 = \sum_{i=1}^\infty (z'_i)^2 = 2$$ and $$\sum_{i=1}^\infty x'_i y'_i = \sum_{i=1}^\infty x'_i z'_i = \sum_{i=1}^\infty y'_i z'_i = 1$$ as inputs, but this time $$\sum_{i=1}^\infty x'_i y'_i z'_i = 1$$
You may find it instructive to read about moments.