I have the following function (where W denotes displacement of cylindrical shell under some force):
$S_w(x_1, x_2) = W(x_1, x_2)W(x_1, x_2)^* = \sum_{m_1, m_2 = 1}^{\infty}C(m_1, m_2) \cdot\sin^2{\frac{m_1 \pi x_1}{L}} \cos^2{\frac{m_2x_2}{R}}$
Where $C$ is some real valued coefficient depending on $m_1$ and $m_2$ (integers). What I need to do is to find the Fourier transform of W defined as follows:
$F[W] = \hat W(k_1, k_2) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W(x_1, x_2)e^{-ik_1x_1}e^{-ik_2x_2}dx_1dx_2$
The problem is I do not have $W(x_1, x_2)$, only $WW^{*}$. For sure, I can compute $F[|W|^2] = F[WW^*]$. But is it possilble to find F[W]?
P.S. My thesis adviser says it is possible to find F[W] using Parseval's or Plancherel's theorem.
P.P.S There is a feeling $W(x_1, x_2) = \sum_{m_1, m_2 = 1}^{\infty}\sqrt C \cdot\sin{\frac{m_1 \pi x_1}{L}} \cos{\frac{m_2x_2}{R}}$ (if we forget about phase factor)