Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum

In the settings of The Real Numbers: Standing Notation and Background and The Flat Torus: Standing Notation, let $M\in\mathbb{N}$, and let the side $L$, the cube $\Gamma_{M}$, the maps $\psi_{k}$, the lattice torus $\mathbb{L}_{M}$, the lattice fields and their real vector space $\mathrm{Map}(\mathbb{L}_{M},\mathbb{R})$, the natural power $L^{n}$ and its multiplicative inverse $L^{-n}$, and the discrete Fourier coefficients $\hat{f}(k)$ of a lattice field $f$ be those of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field. $\mathrm{Map}(\Gamma_{M},\mathbb{R})$ is the real vector space of maps $\Gamma_{M}\to\mathbb{R}$, and $\mathrm{Map}(\mathbb{L}_{M},\{-1,1\})$ is the set of maps $\mathbb{L}_{M}\to\{-1,1\}$; here $\{-1,1\}=\{-1\}\cup\{1\}$ is a nonempty finite set by claim 2 of Basic Properties of Finite Sets and claim 1 of Peeling an Element off a Finite Set, and Unions of Finite Sets, so $\mathrm{Map}(\mathbb{L}_{M},\{-1,1\})$ is a nonempty finite set by The Maps from a Finite Set to a Nonempty Finite Set Form a Nonempty Finite Set §finite. Sums over finite index sets are those of Sum over a Finite Index Set. For $k,l\in\Gamma_{M}$ put $\delta_{kl}=1$ if $k=l$ and $\delta_{kl}=0$ otherwise. $\exp$ is the exponential function and $\log$ the natural logarithm; for every $s\in\mathbb{R}$ the number $\exp(s)+\exp(-s)$ is positive, by claim 2 of Basic Properties of the Exponential Function and since a sum of two positive real numbers is positive, so its logarithm is defined.

1. (Counting the sites)¶ $\displaystyle\sum_{z\in\mathbb{L}_{M}}1=L^{n}$.

2. (Counting the modes)¶ $\displaystyle\sum_{k\in\Gamma_{M}}1=L^{n}$.

3. (Orthogonality)¶ For all $k,l\in\Gamma_{M}$,

$$L^{-n}\sum_{z\in\mathbb{L}_{M}}\psi_{k}(z)\,\psi_{l}(z)=\delta_{kl}.$$

4. (Inversion formula)¶ For every lattice field $f$ and every $z\in\mathbb{L}_{M}$,

$$f(z)=\sum_{k\in\Gamma_{M}}\hat{f}(k)\,\psi_{k}(z).$$

5. (The transform is a linear bijection)¶ The map $f\mapsto\hat{f}$ from $\mathrm{Map}(\mathbb{L}_{M},\mathbb{R})$ to $\mathrm{Map}(\Gamma_{M},\mathbb{R})$ is linear and a bijection, and its inverse sends $b\in\mathrm{Map}(\Gamma_{M},\mathbb{R})$ to the lattice field $z\mapsto\sum_{k\in\Gamma_{M}}b(k)\,\psi_{k}(z)$.

6. (Parseval)¶ For all lattice fields $f$ and $g$,

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

7. (The spin sum)¶ For every lattice field $t$,

$$\sum_{\sigma\in\mathrm{Map}(\mathbb{L}_{M},\{-1,1\})}\exp\Bigl(\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,t(z)\Bigr)=\exp\Bigl(\sum_{z\in\mathbb{L}_{M}}\log\bigl(\exp(t(z))+\exp(-t(z))\bigr)\Bigr).$$