How to calculate a numerical multidimensional Fourier series

Question

The Fourier series tutorial here describes a number of different commands for numerically finding Fourier series and their coefficients. However, they only seem to work for one dimensional Fourier series. Is there an extension which includes multidimensional Fourier series? Specifically, I'm interested in being able to calculate a number of two dimensional Fourier series coefficients numerically.

As an example, find the Fourier coefficients of

$$ \ln(4(\cos(\theta)-\cos(\phi))^{2}+(\sin(\theta)-\sin(\phi))^{2}) $$

on the rectangle $0<\theta,\phi<2\pi$.

Thanks in advance for any help.

Answer 1

For your original example, TrigToExp does what you need:

4 (Cos[θ] - Cos[φ])^2 + (Sin[θ] - Sin[φ])^2 // TrigToExp

(*    5 + 3/4 E^(-2 I θ) + 3/4 E^(2 I θ) - 3/2 E^(-I θ - I φ) -
      5/2 E^(I θ - I φ) - 5/2 E^(-I θ + I φ) - 3/2 E^(I θ + I φ) +
      3/4 E^(-2 I φ) + 3/4 E^(2 I φ)                                  *)

For more complicated functions without an explicit exponential structure, you can evaluate the Fourier integrals explicitly:

f[θ_, φ_] = Log[4 (Cos[θ] - Cos[φ])^2 + (Sin[θ] - Sin[φ])^2 + 1];

F[a_?NumericQ, b_?NumericQ] := 1/(2 π)^2 * NIntegrate[
    f[θ, φ] * E^(-I*{a, b}.{θ, φ}), {θ, 0, 2 π}, {φ, 0, 2 π}]

For example, sum up all Fourier components up to order 2:

g[θ_, φ_] = Sum[F[a, b]*E^(I*{a, b}.{θ, φ}), {a, -2, 2}, {b, -2, 2}] // Chop

(*    1.48242 +
      0.0545782 E^(-2 I θ) +
      0.0545782 E^(2 I θ) - 
      0.0291498 E^(I (-2 θ - 2 φ)) - 
      0.136866 E^(I (2 θ - 2 φ)) - 
      0.224188 E^(I (-θ - φ)) - 
      0.517258 E^(I (θ - φ)) +
      0.0545782 E^(-2 I φ) + 
      0.0545782 E^(2 I φ) -
      0.517258 E^(I (-θ + φ)) - 
      0.224188 E^(I (θ + φ)) - 
      0.136866 E^(I (-2 θ + 2 φ)) - 
      0.0291498 E^(I (2 θ + 2 φ))    *)

Plot3D[{f[θ, φ], Re[g[θ, φ]]}, {θ, 0, 2 π}, {φ, 0, 2 π}]