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

Each result cited is universally quantified over the data in its own statement. Two coefficient families are equal exactly when their values agree at every $k\in\mathbb{Z}^{n}$, and the pointwise operations of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ are those of The Real Vector Space of Real-Valued Functions on a Set §vector-space. The inner-product norm of $L^{2}(\mathbb{T}^{n})$ is $\lVert\,\cdot\,\rVert_{L^{2}}$ by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, and differences in it are those of Elementary Identities in a Vector Space. The inner product identities of Elementary Identities in a Real Inner Product Space §bilinear and the conditions of Real Inner Product Space §inner-product are used for $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$. For an enumeration $\kappa$, the sequence $(E_{\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal basis of $L^{2}(\mathbb{T}^{n})$ by The Trigonometric System is an Orthonormal Basis of the Square-Integrable Space of the Torus §basis, so that Orthonormal Expansions in a Real Hilbert Space, in the setting of Real Hilbert Spaces: Standing Notation and Background, applies to $H=L^{2}(\mathbb{T}^{n})$ with $e_{j}=E_{\kappa(j)}$.

Claim 1. Let $U,U'\in L^{2}(\mathbb{T}^{n})$, $t\in\mathbb{R}$ and $k\in\mathbb{Z}^{n}$. By condition (b) of Real Inner Product Space §inner-product, $\widehat{U+U'}(k)=\langle U+U',E_{k}\rangle_{L^{2}}=\langle U,E_{k}\rangle_{L^{2}}+\langle U',E_{k}\rangle_{L^{2}}=\hat{U}(k)+\hat{U}'(k)=(\hat{U}+\hat{U}')(k)$, and by condition (c), $\widehat{tU}(k)=t\langle U,E_{k}\rangle_{L^{2}}=(t\hat{U})(k)$. As $k$ was arbitrary, $\widehat{U+U'}=\hat{U}+\hat{U}'$ and $\widehat{tU}=t\hat{U}$, which are the two conditions of Linear Map.

Claim 2. Let $\hat{U}=\hat{U}'$. For every $k\in\mathbb{Z}^{n}$, Elementary Identities in a Real Inner Product Space §bilinear gives $\langle U-U',E_{k}\rangle_{L^{2}}=\langle U,E_{k}\rangle_{L^{2}}-\langle U',E_{k}\rangle_{L^{2}}=\hat{U}(k)-\hat{U}'(k)=0$. By The Trigonometric System is an Orthonormal Basis of the Square-Integrable Space of the Torus §complete, $U-U'=0_{L^{2}}$, and adding $U'$ to both sides, using the conditions of Vector Space over a Field and claim 2 of Elementary Identities in a Vector Space, gives $U=U'$.

Claim 3. $\hat{E}_{k}(k')=\langle E_{k},E_{k'}\rangle_{L^{2}}$, which equals $1$ if $k=k'$ and $0$ otherwise by The Trigonometric System on the Torus is Orthonormal §orthonormal.

Claim 4. Let $\kappa$ be an enumeration and $U,U'\in L^{2}(\mathbb{T}^{n})$. Since $\langle U,E_{\kappa(j)}\rangle_{L^{2}}=\hat{U}(\kappa(j))$, Orthonormal Expansions in a Real Hilbert Space §parseval, applied with $x=U$ and $y=U'$, states that $\sum_{j=1}^{\infty}\hat{U}(\kappa(j))\hat{U}'(\kappa(j))$ converges with sum $\langle U,U'\rangle_{L^{2}}$ and that $\sum_{j=1}^{\infty}\hat{U}(\kappa(j))^{2}$ has sum $\langle U,U\rangle_{L^{2}}=(\lVert U\rVert_{L^{2}})^{2}$, the last equality by Real Inner Product Space §norm; and Orthonormal Expansions in a Real Hilbert Space §expansion states that $\sum_{j=1}^{\infty}\hat{U}(\kappa(j))E_{\kappa(j)}$ converges in $L^{2}(\mathbb{T}^{n})$ with sum $U$.

Claim 5. Let $\kappa$ be an enumeration and $c\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ with $\sum_{j=1}^{\infty}c(\kappa(j))^{2}$ convergent. Applying Orthonormal Expansions in a Real Hilbert Space §riesz-fischer to the sequence $c_{j}=c(\kappa(j))$ of real numbers, the series $\sum_{j=1}^{\infty}c(\kappa(j))E_{\kappa(j)}$ converges in $L^{2}(\mathbb{T}^{n})$; writing $U$ for its sum, $(\lVert U\rVert_{L^{2}})^{2}=\sum_{j=1}^{\infty}c(\kappa(j))^{2}$ and $\langle U,E_{\kappa(j)}\rangle_{L^{2}}=c(\kappa(j))$ for every $j\in\mathbb{N}$. Given $k\in\mathbb{Z}^{n}$, by Bijection of Sets there is $j\in\mathbb{N}$ with $\kappa(j)=k$, so $\hat{U}(k)=c(k)$; hence $\hat{U}=c$. If $U'\in L^{2}(\mathbb{T}^{n})$ also satisfies $\hat{U}'=c$, then $U'=U$ by claim 2. This proves claim 5.

Claim 6. Fix $m\in\mathbb{N}$. Since $\rho_{k}\ne0$, claim 4 of Properties of Natural Number Powers in a Field gives $\rho_{k}^{m}\ne0$ for every $k\in\mathbb{Z}^{n}$, so $(\rho_{k}^{m})^{-1}$ exists.

Uniqueness. If $W,W'\in L^{2}(\mathbb{T}^{n})$ both satisfy $\hat{W}(k)=\rho_{k}^{m}c(k)=\hat{W}'(k)$ for every $k$, then $\hat{W}=\hat{W}'$ and $W=W'$ by claim 2.

Subspace. By Elementary Identities in a Real Inner Product Space §zero, $\widehat{0_{L^{2}}}(k)=\langle0_{L^{2}},E_{k}\rangle_{L^{2}}=0=\rho_{k}^{m}\cdot0$ for every $k$, so the zero family lies in $\mathcal{H}_{m}$, with $\Lambda_{m}$ of it equal to $0_{L^{2}}$. Let $c,d\in\mathcal{H}_{m}$, $t\in\mathbb{R}$, $W=\Lambda_{m}c$ and $W'=\Lambda_{m}d$. By claim 1 and distributivity in $\mathbb{R}$, for every $k$,

so $c+d$ and $tc$ lie in $\mathcal{H}_{m}$, with $\Lambda_{m}(c+d)=W+W'=\Lambda_{m}c+\Lambda_{m}d$ and $\Lambda_{m}(tc)=tW=t\Lambda_{m}c$ by uniqueness. Thus $\mathcal{H}_{m}$ satisfies the three conditions of Linear Subspace, so it is a linear subspace of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ and, by The Real Vector Space of Real-Valued Functions on a Set §subspace, a real vector space under the pointwise operations; and $\Lambda_{m}$ satisfies the two conditions of Linear Map for that vector space structure.

Bijection. Let $W\in L^{2}(\mathbb{T}^{n})$. Define $c\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ by $c(k)=(\rho_{k}^{m})^{-1}\hat{W}(k)$; then $\rho_{k}^{m}c(k)=\hat{W}(k)$ for every $k$, so $c\in\mathcal{H}_{m}$ and $\Lambda_{m}c=W$. If $d\in\mathcal{H}_{m}$ also satisfies $\Lambda_{m}d=W$, then $\rho_{k}^{m}d(k)=\hat{W}(k)=\rho_{k}^{m}c(k)$ for every $k$, and multiplying by $(\rho_{k}^{m})^{-1}$ gives $d(k)=c(k)$, so $d=c$. Hence every $W$ has exactly one preimage under $\Lambda_{m}$, which is the condition of Bijection of Sets.