The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field

In the settings of The Real Numbers: Standing Notation and Background and The Flat Torus: Standing Notation, with the dimension $n$ and the initial segment $[n]$ of The Flat Torus: Standing Notation §background, and the integer lattice $\mathbb{Z}^{n}$ of The Flat Torus: Standing Notation §cell. The map written $e_{k}$ in the trigonometric system on $\mathbb{T}^{n}$ is written $\psi_{k}:\mathbb{R}^{n}\to\mathbb{R}$ ($k\in\mathbb{Z}^{n}$) here. Finite sets are those of Finite Set, and sums over finite index sets those of Sum over a Finite Index Set. Let $M\in\mathbb{N}$ (the cutoff), put $L=2M+1\in\mathbb{N}$ (the side), a natural number and hence positive and invertible in $\mathbb{R}$ by The Real Numbers: Standing Notation and Background §numbers, and let $\Gamma_{M}$ be the cube of The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points, the set of the $k\in\mathbb{Z}^{n}$ with $-M\le k_{i}\le M$ for every $i\in[n]$; it is a nonempty finite set by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite.

1. (The lattice torus)¶ $W_{L}$ is the set of the $w\in\mathbb{Z}^{n}$ with $0\le w_{i}\le L-1$ for every $i\in[n]$. It contains the point all of whose components are $0$, and it is a subset of the cube $\Gamma_{2M}$, since $L-1=2M$; so it is finite by claim 3 of Basic Properties of Finite Sets and The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite. The lattice torus of side $L$ is the set

$$\mathbb{L}_{M}=\{L^{-1}w:w\in W_{L}\}\subseteq\mathbb{R}^{n},$$

where $L^{-1}w$ is the scalar multiple of $w$ by the multiplicative inverse $L^{-1}$ of $L$. It is the image of the nonempty finite set $W_{L}$ under the map $w\mapsto L^{-1}w$, so it is nonempty and finite by claim 4 of Basic Properties of Finite Sets. Its elements are the sites.

2. (Lattice fields)¶ A lattice field is a map $f:\mathbb{L}_{M}\to\mathbb{R}$. The set $\mathrm{Map}(\mathbb{L}_{M},\mathbb{R})$ of lattice fields, with pointwise addition and multiplication by scalars, is a real vector space.

3. (Discrete Fourier coefficients)¶ The discrete Fourier coefficients of a lattice field $f$ are the real numbers

$$\hat{f}(k)=L^{-n}\sum_{z\in\mathbb{L}_{M}}f(z)\,\psi_{k}(z)\qquad(k\in\Gamma_{M}),$$

where $L^{-n}$ is the multiplicative inverse of the natural power $L^{n}$, which is positive: it is nonnegative by Properties of Natural Number Powers in a Field §nonnegative and nonzero by Properties of Natural Number Powers in a Field §vanishing, $L$ being positive. The map $\hat{f}:\Gamma_{M}\to\mathbb{R}$, an element of the real vector space $\mathrm{Map}(\Gamma_{M},\mathbb{R})$, is the discrete Fourier transform of $f$.