Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$, and in the setting of Real Hilbert Spaces: Standing Notation and Background, whose standing space $H$ is not used here; the integer lattice $\mathbb{Z}^{n}$, Euclidean space $\mathbb{R}^{n}$ with its norm $\lVert\,\cdot\,\rVert$, and the real Hilbert space $L^{2}(\mathbb{T}^{n})$ are the ones fixed there, with inner product $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ and norm $\lVert\,\cdot\,\rVert_{L^{2}}$ and zero vector $0_{L^{2}}$. Let $E_{k}$ for $k\in\mathbb{Z}^{n}$ be the classes of the trigonometric system introduced in The Trigonometric System on the Torus is Orthonormal (i), and for $U\in L^{2}(\mathbb{T}^{n})$ let $\hat{U}$ be its Fourier coefficient family, an element of the set $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ of all maps from $\mathbb{Z}^{n}$ to $\mathbb{R}$, which is a real vector space under the pointwise operations, with zero vector the zero family, the map with value $0$ everywhere; its elements are called coefficient families. A bijection $\kappa:\mathbb{N}\to\mathbb{Z}^{n}$ is called an enumeration of the lattice; one exists by The Integer Lattice Admits an Enumeration by the Natural Numbers (ii). Series of real numbers and series in $L^{2}(\mathbb{T}^{n})$ and their sums are as defined there. Powers $t^{m}$ of a real number $t$ with $m\in\mathbb{N}$ are natural powers; in particular $t^{2}=tt$, with $2=1+1$, by claim 1 of Properties of Natural Number Powers in a Field.

Let $\mu_{k}$ for $k\in\mathbb{Z}^{n}$ be the Fourier weights fixed there, that lemma being used with $s=n$; they are positive by Summability of the Negative Powers of the Fourier Weights of the Torus (i), so that $\tfrac{1}{\mu_{k}}$ is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. For $k\in\mathbb{Z}^{n}$ let $\rho_{k}$ be the nonnegative real number with $\rho_{k}^{2}=\tfrac{1}{\mu_{k}}$ furnished by Existence and Uniqueness of the Nonnegative Square Root; it is nonzero, since $0\cdot0=0$ by claim 1 of Zero Products and Elementary Identities in a Field, hence positive. Then the following hold.

1. (Linearity)(i) For all $U,U'\in L^{2}(\mathbb{T}^{n})$ and $t\in\mathbb{R}$, $\widehat{U+U'}=\hat{U}+\hat{U}'$ and $\widehat{tU}=t\hat{U}$; that is, the map $U\mapsto\hat{U}$ from $L^{2}(\mathbb{T}^{n})$ to $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ is linear.

2. (Injectivity)(ii) If $U,U'\in L^{2}(\mathbb{T}^{n})$ satisfy $\hat{U}=\hat{U}'$, then $U=U'$.

3. (The trigonometric classes)(iii) For $k,k'\in\mathbb{Z}^{n}$, $\hat{E}_{k}(k')$ equals $1$ if $k'=k$ and equals $0$ if $k'\ne k$.

4. (Parseval's identity and the Fourier expansion)(iv) Let $\kappa$ be an enumeration and $U,U'\in L^{2}(\mathbb{T}^{n})$. Then the series $\sum_{j=1}^{\infty}\hat{U}(\kappa(j))\,\hat{U}'(\kappa(j))$ converges with sum $\langle U,U'\rangle_{L^{2}}$, in particular

$$\sum_{j=1}^{\infty}\hat{U}(\kappa(j))^{2}=\bigl(\lVert U\rVert_{L^{2}}\bigr)^{2},$$

and the series $\sum_{j=1}^{\infty}\hat{U}(\kappa(j))\,E_{\kappa(j)}$ converges in $L^{2}(\mathbb{T}^{n})$ with sum $U$.

5. (Square-summable families are Fourier coefficient families)(v) Let $\kappa$ be an enumeration and let $c\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ be such that the series $\sum_{j=1}^{\infty}c(\kappa(j))^{2}$ converges. Then there is exactly one $U\in L^{2}(\mathbb{T}^{n})$ with $\hat{U}=c$; it is the sum of the series $\sum_{j=1}^{\infty}c(\kappa(j))\,E_{\kappa(j)}$, which converges in $L^{2}(\mathbb{T}^{n})$, and $(\lVert U\rVert_{L^{2}})^{2}=\sum_{j=1}^{\infty}c(\kappa(j))^{2}$.

6. (Realisation of weighted coefficient families)(vi) Let $m\in\mathbb{N}$, and let $\mathcal{H}_{m}$ be the set of those $c\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ for which there is $W\in L^{2}(\mathbb{T}^{n})$ with

$$\hat{W}(k)=\rho_{k}^{m}\,c(k)\qquad\text{for every }k\in\mathbb{Z}^{n}.$$

Then for every $c\in\mathcal{H}_{m}$ there is exactly one such $W$, written $\Lambda_{m}c$; the set $\mathcal{H}_{m}$ is a linear subspace of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$, hence a real vector space under the pointwise operations; and the map $\Lambda_{m}:\mathcal{H}_{m}\to L^{2}(\mathbb{T}^{n})$ is linear and a bijection.