Form of fourier series in 3D coordinate

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I have an interesting problem where a constrained solver is used to estimate a periodic function as part of a sensor calibration process.

The idea is to write this periodic function in the form of Fourier Series and let the constrained solver do all the hard work. I cannot use a Fourier transform since the goal function must be updated as the data comes in.

For example, a 1D version of this would be modelling the goal function as:

$$f(x) = A + B_1\cos(\text{angle} + C_1) + B_2\cos(\text{angle} + C_2) + B_3\cos(\text{angle} + C_3)$$

(stopping at 3 cos functions is... trial and error).

So what would be the equivalent in 3D, i.e $f(x,y,z)$?