In usual Fourier analysis, we have a periodic function like $f: \mathbb{R}^3 \rightarrow \mathbb{C}$. The function can be decomposed into a countable basis (i.e. $\{\mathrm{e}^{i\mathbf{k_n}\cdot \mathbf{x}} \mid n\in \mathbb{Z}\}$) and the coefficients easily calculated by $c_n=\int_V f(\mathbf{x}) \mathrm{e}^{i\mathbf{k}_n\cdot \mathbf{x}} d\mathbf{x}$.
But what about when we have a periodic function $g: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ which is a vector field? What's the countable set of basis functions in this case and how can one get the coefficients of the Fourier series for $g$?
I would also appreciate any resources that discuss this topic.

  • $\begingroup$ The Hilbert space $L^2(\Bbb{R^3/Z^3, R^d})$ of square-integrable functions $\Bbb{R^3/Z^3 \to R^d}$ with inner product $\langle f,g \rangle = \iiint_{[0,1]^3} (f(x),g(x))d^3 x$ (where $(.,.)$ is the usual inner product on $\Bbb{R}^d$) is just the direct sum of d copies of the Hilbert space $L^2(\Bbb{R^3/Z^3,R})$. Thus a natural orthonormal basis is... $\endgroup$ – reuns May 9 at 4:03

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