The Fourier Coefficients of a Square-Integrable Class on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the integer lattice $\mathbb{Z}^{n}$ and the real Hilbert space $L^{2}(\mathbb{T}^{n})$ are the ones fixed there, $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ is its inner product, and $E_{k}$ for $k\in\mathbb{Z}^{n}$ are the classes of the trigonometric system introduced in The Trigonometric System on the Torus is Orthonormal (i). Let $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ be the set of all maps from $\mathbb{Z}^{n}$ to $\mathbb{R}$, as in The Real Vector Space of Real-Valued Functions on a Set.

(i) Let $U\in L^{2}(\mathbb{T}^{n})$. The Fourier coefficient family of $U$, with respect to the real trigonometric system, is the map $\hat{U}\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ given by

$$\hat{U}(k)=\langle U,E_{k}\rangle_{L^{2}}\qquad(k\in\mathbb{Z}^{n}),$$

and $\hat{U}(k)$ is called the $k$th Fourier coefficient of $U$.