\renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\dag}{\dagger} \newcommand{\const}{\mathrm{const}} \newcommand{\arcsinh}{\operatorname{arcsinh}}
Let function u(\mathbf{x}), \mathbf{x}=(x_1,x_2,\ldots,x_n) be 2\pi-periodic with respect to each variable x_1,x_2,\ldots,x_n. Then \begin{equation} u(\mathbf{x})= \sum_{\mathbf{m}\in \mathbb{Z}^n} c_{\mathbf{m}} e^{ i \mathbf{m}\cdot \mathbf{x}} \label{eq-4.B.1} \end{equation} with \begin{equation} c_{\mathbf{m}} =(2\pi)^{-n} \iiint_\Omega e^{- i \mathbf{m}\cdot \mathbf{x}} u(\mathbf{x})\,d^nx \label{eq-4.B.2} \end{equation} and \begin{equation} \sum_{\mathbf{m}\in \mathbb{Z}^n} |c_{\mathbf{m}}|^2 =(2\pi)^{-n} \iiint_\Omega |u(\mathbf{x})|^2\,d^n x \label{eq-4.B.3} \end{equation} where \Omega=(0,2\pi)^n is n-dimensional cube. Here and below we write n-dimensional integral as \iiint.
Those formulae follows by repetition of what we have in 1-dimensional case: we simply make Fourier series decomposition by x_1, then by x_2 and so on ... ; for example, for n=2 \begin{align*} &u(x,y) =\sum_{m\in \mathbb{Z}} u_m(y)e^{ixm}, &&u_m(y)=(2\pi)^{-1}\int_0^{2\pi}u(x,y)e^{-imx}\,dx,\\ &u_m(y)=\sum_{m\in \mathbb{Z}} c_{mn} e^{iny}, && c_{mn}=(2\pi)^{-1}\int_0^{2\pi}u_m(y)e^{-iny}\,dy\\ \implies &u(x,y)=\sum_{(m,n)\in \mathbb{Z}^2}c_{mn} e^{imx+iny}, && c_{mn}=(2\pi)^{-2}\iint_{[0,2\pi]^2} u (x,y) e^{-imx-iny}\,dxdy. \end{align*}
We need slightly generalize these formulae.
Definition 1. Let \Gamma be n-dimentional lattice. It means that there are n linearly independent vectors \mathbf{e}_1, \ldots, \mathbf{e}_n and \begin{equation} \Gamma = \{ k_1 \mathbf{e}_1 + k_2 \mathbf{e}_2+\ldots +k_n \mathbf{e}_n:\, k_1,k_2,\ldots,k_n\in \mathbb{Z}\} \label{eq-4.B.4} \end{equation}
Remark 1. The same lattice \Gamma is defined by vectors \mathbf{e}'_1, \ldots, \mathbf{e}'_n with \mathbf{e}'_j=\sum _k \alpha_{jk}\mathbf{e}_k with integer coefficients if and only if the determinant of the matrix (\alpha_{jk})_{j,k=1,\ldots,n} of coefficients is equal \pm 1.
Definition 2. Let \Gamma be n-dimentional lattice. We call u(\mathbf{x}) periodic with respect to \Gamma or simply \Gamma-periodic if \begin{equation} u(\mathbf{x}+\mathbf{y})= u(\mathbf{x})\qquad \forall \mathbf{y}\in \Gamma\ \forall \mathbf{x}. \label{eq-4.B.5} \end{equation}
In the previous Subsection \Gamma= (2\pi\mathbb{Z})^n. Let us change coordinate system so that \Gamma becomes (2\pi\mathbb{Z})^n, apply (\ref{eq-4.B.1})--(\ref{eq-4.B.3}) and then change coordinate system back. We get \begin{equation} u(\mathbf{x})= \sum_{\mathbf{m}\in \Gamma^*} c_{\mathbf{m}} e^{ i \mathbf{m}\cdot \mathbf{x}} \label{eq-4.B.6} \end{equation} with \begin{equation} c_{\mathbf{m}} =|\Omega |^{-1} \iiint_\Omega e^{- i \mathbf{m}\cdot \mathbf{x}} u(\mathbf{x})\,d^n x \label{eq-4.B.7} \end{equation} and \begin{equation} \sum_{\mathbf{m}\in \Gamma^*} |c_{\mathbf{m}}|^2 =|\Omega|^{-1} \iiint_\Omega |u(\mathbf{x})|^2\,d^n x \label{eq-4.B.8} \end{equation} where |\Omega| is a volume of \Omega and
Definition 3.
Remark 2. We prefer to use original coordinate system rather than one with coordinate vectors (2\pi)^{-1}\mathbf{e}_1,\ldots, (2\pi)^{-1}\mathbf{e}_n because the latter is not necessarily orthonormal and in it Laplacian will have a different form.
These notions are important for studying the band spectrum of the Schrödinger operator -\Delta +V(\mathbf{x}) with periodic (with respect to some lattice \Gamma) potential in the whole space which has applications to the Physics of crystals. For this the following decomposition is used for functions u(\mathbf{x}) in the whole space \mathbb{R}^n.
Theorem 1. Let u(\mathbf{x}) be sufficiently fast decaying function on \mathbb{R}^n. Then \begin{equation} u(\mathbf{x})= \iiint_{\Omega^*} u(\mathbf{k};\mathbf{x})\,d^n\mathbf{k} \label{eq-4.B.10} \end{equation} with \begin{equation} u(\mathbf{k};\mathbf{x})= (2\pi)^{-n}|\Omega| \sum_{\mathbf{l}\in \Gamma} e^{-i\mathbf{k}\cdot \mathbf{l}} u(\mathbf{x}+\mathbf{l}). \label{eq-4.B.11} \end{equation} Here u(\mathbf{k};\mathbf{x}) is quasiperiodic with quasimomentum \mathbf{k} \begin{equation} u(\mathbf{k};\mathbf{x}+\mathbf{y})= e^{i\mathbf{k}\cdot\mathbf{y}}u(\mathbf{k};\mathbf{x})\qquad \forall \mathbf{y}\in \Gamma\ \forall \mathbf{x}. \label{eq-4.B.12} \end{equation}
Lattice and elementary cell (teal) and dual lattice and dual elementary cell (orange)
Proof. Observe that since u is sufficiently fast decaying series in (\ref{eq-4.B.11}) converges and one can see easily that it defines quasiperiodic with quasimomentum \mathbf{k} function.
The proof of (\ref{eq-4.B.10}) is trivial because \begin{gather*} \iiint_{\Omega^*} e^{-i\mathbf{k}\cdot \mathbf{l}} \,d^n\mathbf{k}= \left\{\begin{aligned} &|\Omega^*| &&\mathbf{l}=0,\\ &0 &&0\ne \mathbf{l}\in \Gamma \end{aligned}\right. \end{gather*} and |\Omega^*|=(2\pi)^n|\Omega|^{-1}. and |\Omega^*|=(2\pi)^n|\Omega|^{-1}.
Remark 3. Since choice of vectors \mathbf{e}_1, \ldots, \mathbf{e}_n for given lattice \Gamma is not unique, the choice of \Omega and \Omega^* is not unique either, but |\Omega| is defined uniquely.