I am hoping to get some clarifications/help on dealing with coefficients of a multi-dimensional Fourier series.

First, I apologize for any mistakes or notations that may be off, I know just enough about Fourier analysis to get by.

I am trying to model an n-dimentional function as a Fourier series like so (will use 3-d just for brevity):

$$ f( \alpha, \beta, \gamma ) = \sum_{p=0}^{\infty} \sum_{q=0}^{\infty} \sum_{r=0}^{\infty} \hat{C_{pq}^{r}} e^{i \alpha p 2\pi t/T}e^{i \beta q 2\pi t/T}e^{i \gamma r 2\pi t/T}$$

This function is discrete and each of the exponentials are independent, thus I can treat each function as a 1-d transform, if I understand correctly, and get coefficients for each individual function $\hat{X_{p}}$, $\hat{Y_{q}}$ and $\hat{Z_{r}}$ respectively.

After reading this post (Identifying the product of two Fourier series with a third?) (and if I also understand that correctly), when finding the coefficients $\hat{C_{pq}^{r}}$ can I simply do the convolution of the coefficients $\hat{C_{pq}^{r}}$ = $\hat{Z_{r}}\ast(\hat{X_{p}}\ast \hat{Y_{q}})$? Or is that bad math?

For example, if I want to find the coefficient $\hat{C_{10}^{2}}$ can I do $\hat{Z_{2}}\ast(\hat{X_{1}}\ast \hat{Y_{0}})$?

I hope I explained that well enough, thanks!


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