Proof of The Laplacian of the Trigonometric System, and the Fourier Coefficients of a Laplacian on the Torus

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named. For $k\in\mathbb{Z}^{n}$ and $i\in[n]$ let $k^{(i)}$ be the point of $\mathbb{R}^{n}$ whose $i$-th component is $-k_{i}$ and whose $j$-th component is $k_{j}$ for $j\in[n]$ with $j\ne i$, as introduced in The Trigonometric System is Continuously Differentiable and Orthogonal in the Sobolev Space of the Torus, which records that $k^{(i)}\in\mathbb{Z}^{n}$ and that

$$e_{k}\in C^{1}_{\mathrm{per}},\qquad \partial_{i}e_{k}(x)=-2\pi k_{i}\,e_{k^{(i)}}(x)\quad\text{for every }x\in\mathbb{R}^{n},$$

by The Trigonometric System is Continuously Differentiable and Orthogonal in the Sobolev Space of the Torus (i). Pointwise sums, scalar multiples and products of maps on $\mathbb{R}^{n}$ are those of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions.

Claim 1 (Reflecting one component twice). Let $k\in\mathbb{Z}^{n}$ and $i\in[n]$. Then $(k^{(i)})_{i}=-k_{i}$ and $(k^{(i)})^{(i)}=k$.

The first is the definition of $k^{(i)}$. For the second, the $i$-th component of $(k^{(i)})^{(i)}$ is $-(k^{(i)})_{i}=-(-k_{i})$, which is $k_{i}$ because in a field the additive inverse of $-k_{i}$ is unique and $k_{i}$ is one such; and for $j\ne i$ the $j$-th component of $(k^{(i)})^{(i)}$ is $(k^{(i)})_{j}=k_{j}$. The two points of $\mathbb{R}^{n}$ therefore have the same components, so they are equal by claim 1 of Euclidean Points as Tuples of Real Numbers.

Claim 2 (Clause 1). Let $k\in\mathbb{Z}^{n}$ and $i\in[n]$. Applying the displayed identity of the preamble to $k^{(i)}$ in place of $k$, and then claim 1,

$$\partial_{i}e_{k^{(i)}}(x)=-2\pi\,(k^{(i)})_{i}\,e_{(k^{(i)})^{(i)}}(x)=-2\pi(-k_{i})\,e_{k}(x)=2\pi k_{i}\,e_{k}(x)$$

for every $x\in\mathbb{R}^{n}$, using claim 2 of Zero Products and Elementary Identities in a Field for $-2\pi(-k_{i})=2\pi k_{i}$.

Now $\partial_{i}e_{k}$ is the scalar multiple $(-2\pi k_{i})e_{k^{(i)}}$ of $e_{k^{(i)}}$, by the preamble. Since the partial derivative of $e_{k^{(i)}}$ with respect to the $i$-th variable exists at every point, claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions gives, for every $x\in\mathbb{R}^{n}$,

$$\partial_{i}\partial_{i}e_{k}(x)=(-2\pi k_{i})\,\partial_{i}e_{k^{(i)}}(x)=(-2\pi k_{i})(2\pi k_{i})\,e_{k}(x)=-4\pi^{2}k_{i}^{2}\,e_{k}(x),$$

the last equality being field arithmetic together with claim 2 of Zero Products and Elementary Identities in a Field.

It remains to see that $e_{k}$ is of class $C^{2}$ on $\mathbb{R}^{n}$ and lies in $C^{2}_{\mathrm{per}}$. It is of class $C^{1}$ on $\mathbb{R}^{n}$, being a member of $C^{1}_{\mathrm{per}}$ by the preamble and Lattice-Periodic Functions and the Periodic Function Classes (iii); and for each $i\in[n]$ the function $\partial_{i}e_{k}=(-2\pi k_{i})e_{k^{(i)}}$ is of class $C^{1}$ on $\mathbb{R}^{n}$ by claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, since $e_{k^{(i)}}$ is. By clause 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ, read with the scalar convention of clause 3 there, $e_{k}$ is of class $C^{2}$ on $\mathbb{R}^{n}$; and it is $\mathbb{Z}^{n}$-periodic, being in $C^{1}_{\mathrm{per}}$, so it lies in $C^{2}_{\mathrm{per}}$ by Lattice-Periodic Functions and the Periodic Function Classes (iii).

Claim 3 (Clause 2). Let $k\in\mathbb{Z}^{n}$ and $x\in\mathbb{R}^{n}$. By The Laplacian of a Twice Continuously Differentiable Function (i), applicable since $e_{k}$ is of class $C^{2}$ on $\mathbb{R}^{n}$ by claim 2, and then by claim 2 again,

$$\Delta e_{k}(x)=\sum_{i=1}^{n}\partial_{i}\partial_{i}e_{k}(x)=\sum_{i=1}^{n}\bigl(-4\pi^{2}e_{k}(x)\bigr)k_{i}^{2}=\bigl(-4\pi^{2}e_{k}(x)\bigr)\sum_{i=1}^{n}k_{i}^{2},$$

the last step by the homogeneity of claim 3 of Properties of Finite Sums. By claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square the remaining sum is $\lVert k\rVert^{2}$, which gives clause 2.

Claim 4 (Regularity in clause 3). Let $u\in C^{2}_{\mathrm{per}}$ and $i\in[n]$. Then $u$ is of class $C^{1}$ on $\mathbb{R}^{n}$ by claim 2 of Euclidean Space Is Open in Itself, hence $u\in C^{1}_{\mathrm{per}}$, and $u\in C_{\mathrm{per}}$ by claim 3 of that lemma with Lattice-Periodic Functions and the Periodic Function Classes (iii). By Elementary Properties of Lattice-Periodic Functions (ii) one has $\partial_{i}u\in C^{1}_{\mathrm{per}}$, and then $\partial_{i}\partial_{i}u\in C_{\mathrm{per}}$ by that clause again.

Next, $\Delta u\in C_{\mathrm{per}}$. For $m\in[n]$ let $P_{m}:\mathbb{R}^{n}\to\mathbb{R}$ be the map with $P_{m}(x)=\sum_{i=1}^{m}\partial_{i}\partial_{i}u(x)$. Throughout this paragraph $m+1$ denotes the successor $S(m)$ of $m$, the two being equal by claim 1 of Arithmetic of Addition on the Natural Numbers, so that the clauses cited below, which are phrased with the successor map, apply as written.

Let $T$ be the union of the set of $m\in\mathbb{N}$ with $n<m$ and the set of $m\in[n]$ with $P_{m}\in C_{\mathrm{per}}$. Then $1\in T$: if $n<1$ this is immediate, and otherwise $P_{1}=\partial_{1}\partial_{1}u$ by claim 1 of Properties of Finite Sums, which lies in $C_{\mathrm{per}}$ by the previous paragraph. Let $m\in T$; we show $m+1\in T$. If $n<m+1$ this is immediate, so assume $m+1\in[n]$; then $m\in[n]$, since $m<m+1$ by claim 5 of Properties of the Order on the Natural Numbers, hence $m\le m+1$ by claim 1 of that lemma, and $m+1\le n$, so $m\le n$ by the transitivity of $\le$ in claim 1. Therefore $P_{m}\in C_{\mathrm{per}}$, and by the recursion in claim 1 of Properties of Finite Sums the map $P_{m+1}$ is the pointwise sum $P_{m}+\partial_{m+1}\partial_{m+1}u$, which lies in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions (i). By Principle of Induction for the Natural Numbers with the inductive set $T$ we get $T=\mathbb{N}$, and since $n\in[n]$ this gives $P_{n}\in C_{\mathrm{per}}$. By The Laplacian of a Twice Continuously Differentiable Function (i) the map $\Delta u$ is $P_{n}$.

Consequently $u|_{Q}$ and $(\Delta u)|_{Q}$ are measurable with respect to $\mathcal{B}_{Q}$ and belong to $\mathcal{L}^{2}(\mathbb{T}^{n})$, by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i); the same applies to $(\partial_{i}u)|_{Q}$, $(\partial_{i}\partial_{i}u)|_{Q}$ and $e_{k}|_{Q}$ for every $k\in\mathbb{Z}^{n}$, the last by The Trigonometric System on the Torus is Orthonormal (i).

Claim 5 (Moving one derivative at a time). Let $u\in C^{2}_{\mathrm{per}}$, $k\in\mathbb{Z}^{n}$ and $i\in[n]$. Then

$$\int_{\mathbb{T}^{n}}(\partial_{i}\partial_{i}u)|_{Q}\,\bigl(e_{k}|_{Q}\bigr)\,dx=-4\pi^{2}k_{i}^{2}\int_{\mathbb{T}^{n}}u|_{Q}\,\bigl(e_{k}|_{Q}\bigr)\,dx .$$

By claim 1 of Elementary Properties of the Weak Partial Derivative on the Torus, applied to $u\in C^{1}_{\mathrm{per}}$ and to $\partial_{i}u\in C^{1}_{\mathrm{per}}$ from claim 4, the class $[(\partial_{i}u)|_{Q}]$ is the $i$-th weak partial derivative of $[u|_{Q}]$ in $L^{2}(\mathbb{T}^{n})$, and $[(\partial_{i}\partial_{i}u)|_{Q}]$ is the $i$-th weak partial derivative of $[(\partial_{i}u)|_{Q}]$ in $L^{2}(\mathbb{T}^{n})$.

The map $\partial_{i}e_{k}$ lies in $C^{1}_{\mathrm{per}}$: it is the scalar multiple $(-2\pi k_{i})e_{k^{(i)}}$ of $e_{k^{(i)}}\in C^{1}_{\mathrm{per}}$, so Elementary Properties of Lattice-Periodic Functions (i) applies. Now apply claim 3 of Elementary Properties of the Weak Partial Derivative on the Torus twice: first with the class $[(\partial_{i}u)|_{Q}]$, its weak derivative $[(\partial_{i}\partial_{i}u)|_{Q}]$, the representatives $(\partial_{i}u)|_{Q}$ and $(\partial_{i}\partial_{i}u)|_{Q}$, and the test function $e_{k}\in C^{1}_{\mathrm{per}}$; then with the class $[u|_{Q}]$, its weak derivative $[(\partial_{i}u)|_{Q}]$, the representatives $u|_{Q}$ and $(\partial_{i}u)|_{Q}$, and the test function $\partial_{i}e_{k}$. This gives

$$\int_{\mathbb{T}^{n}}(\partial_{i}u)|_{Q}\,\bigl((\partial_{i}e_{k})|_{Q}\bigr)\,dx=-\int_{\mathbb{T}^{n}}(\partial_{i}\partial_{i}u)|_{Q}\,\bigl(e_{k}|_{Q}\bigr)\,dx$$

and

$$\int_{\mathbb{T}^{n}}u|_{Q}\,\bigl((\partial_{i}\partial_{i}e_{k})|_{Q}\bigr)\,dx=-\int_{\mathbb{T}^{n}}(\partial_{i}u)|_{Q}\,\bigl((\partial_{i}e_{k})|_{Q}\bigr)\,dx .$$

Combining the two, the left-hand side of the first display equals the negative of the left-hand side of the second, so

$$\int_{\mathbb{T}^{n}}(\partial_{i}\partial_{i}u)|_{Q}\,\bigl(e_{k}|_{Q}\bigr)\,dx=\int_{\mathbb{T}^{n}}u|_{Q}\,\bigl((\partial_{i}\partial_{i}e_{k})|_{Q}\bigr)\,dx .$$

By claim 2 the map $\partial_{i}\partial_{i}e_{k}$ is the scalar multiple $(-4\pi^{2}k_{i}^{2})e_{k}$, so the integrand on the right is $(-4\pi^{2}k_{i}^{2})\,u|_{Q}\,(e_{k}|_{Q})$ at every point of $Q$. The product $u|_{Q}\,(e_{k}|_{Q})$ is integrable, being a product of two members of $\mathcal{L}^{2}(\mathbb{T}^{n})$ by claim 4 and L²(μ) Is a Real Hilbert Space (i); so Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions (ii), used with the single summand $u|_{Q}\,(e_{k}|_{Q})$ and the constant $-4\pi^{2}k_{i}^{2}$, moves that constant out of the integral and yields the assertion.

Claim 6 (Clause 3). Let $u\in C^{2}_{\mathrm{per}}$ and $k\in\mathbb{Z}^{n}$; the membership assertions of clause 3 are claim 4. By L²(μ) Is a Real Hilbert Space (i), applied to the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ of The Flat Torus: Standing Notation (iii), both inner products in clause 3 are the integrals over $\mathbb{T}^{n}$ of the corresponding pointwise products.

At every point of $Q$ one has $(\Delta u)|_{Q}\,(e_{k}|_{Q})=\sum_{i=1}^{n}(\partial_{i}\partial_{i}u)|_{Q}\,(e_{k}|_{Q})$, by The Laplacian of a Twice Continuously Differentiable Function (i) and the homogeneity of claim 3 of Properties of Finite Sums. Each summand is integrable, being a product of two members of $\mathcal{L}^{2}(\mathbb{T}^{n})$ by claim 4 and L²(μ) Is a Real Hilbert Space (i); so Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions (ii) gives

$$\int_{\mathbb{T}^{n}}(\Delta u)|_{Q}\,\bigl(e_{k}|_{Q}\bigr)\,dx=\sum_{i=1}^{n}\int_{\mathbb{T}^{n}}(\partial_{i}\partial_{i}u)|_{Q}\,\bigl(e_{k}|_{Q}\bigr)\,dx .$$

By claim 5 each term equals $-4\pi^{2}k_{i}^{2}\int_{\mathbb{T}^{n}}u|_{Q}\,(e_{k}|_{Q})\,dx$, so by the homogeneity of claim 3 of Properties of Finite Sums the right-hand side is

$$\Bigl(-4\pi^{2}\int_{\mathbb{T}^{n}}u|_{Q}\,\bigl(e_{k}|_{Q}\bigr)\,dx\Bigr)\sum_{i=1}^{n}k_{i}^{2},$$

and $\sum_{i=1}^{n}k_{i}^{2}=\lVert k\rVert^{2}$ by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. This is clause 3.