Proof of The Trigonometric System on the Torus is Orthonormal

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. The field axioms of $\mathbb{R}$ are used freely for associativity, commutativity and distributivity, and for $s\cdot1=s$. Transitivity of $\le$ is used freely: if $a\le b$ and $b\le c$ then $0\le b-a$ and $0\le c-b$ by claim 3 of Elementary Arithmetic in an Ordered Field, hence $0\le c-a$ by claim 2 of that lemma, hence $a\le c$; a chain in which one of the two inequalities is strict is handled instead by the mixed transitivity of claim 2 of Elementary Order Arithmetic in an Ordered Field. By Finite Product Notation in a Field a finite product is determined by the map on the initial segment it is formed over, so two such products agree as soon as the two maps agree; this is used without further comment. Throughout, $C_{j}$ and $S_{j}$ for $j\in\mathbb{Z}$ are the trigonometric monomials of Cell Integrals of the Trigonometric Monomials, in terms of which $\phi_{m}$ is defined in The Trigonometric System on the Torus §one-dimensional.

Claim 1. (Elementary properties of $\sqrt{2}$.) $0\le\sqrt{2}$, $\sqrt{2}\,\sqrt{2}=2$, $|\sqrt{2}|=\sqrt{2}$ and $1\le\sqrt{2}$.

Proof of Claim 1. The first two hold by the description of $\sqrt{2}$ in The Trigonometric System on the Torus, since $t^{2}=tt$. The third holds because Absolute Value in an Ordered Field sets $|t|=t$ whenever $0\le t$. For the fourth, $1\le2$ because $0\le1$ by claim 1 of Elementary Arithmetic in an Ordered Field and $2=1+1$, so claim 3 of that lemma applies; since $1\cdot1=1$ and $\sqrt{2}\,\sqrt{2}=2$, and both $1$ and $\sqrt{2}$ are nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field turns $1\le2$ into $1\le\sqrt{2}$. This proves Claim 1.

Claim 2. (The one-variable factors.) Let $j\in\mathbb{Z}$. Then $\phi_{j}$ is continuous on $\mathbb{R}$; $|\phi_{j}(t)|\le\sqrt{2}$ for every $t\in\mathbb{R}$; and $\phi_{j}(t+p)=\phi_{j}(t)$ for every $t\in\mathbb{R}$ and every $p\in\mathbb{Z}$.

Proof of Claim 2. By claim 1 of Arithmetic, Order and Discreteness of the Integers exactly one of $j=0$, $0<j$ and $j<0$ holds.

If $j=0$ then $\phi_{0}$ is the map with constant value $1$. It is continuous at every $x\in\mathbb{R}$ relative to $\mathbb{R}$ by claim 1 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, applied with the metric space $(\mathbb{R},d_{\mathbb{R}})$, with $A=\mathbb{R}$ and with the constant $b=1$, and $x$ is arbitrary, so $\phi_{0}$ is continuous on $\mathbb{R}$; $|\phi_{0}(t)|=|1|=1\le\sqrt{2}$ by Absolute Value in an Ordered Field and Claim 1; and $\phi_{0}(t+p)=1=\phi_{0}(t)$.

Suppose $0<j$, so $\phi_{j}=\sqrt{2}\,C_{j}$. The map $C_{j}$ is continuous on $\mathbb{R}$ by Cell Integrals of the Trigonometric Monomials §calculus, hence so is $\phi_{j}$ by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, in its scalar-multiple form $cf$, applied on $\mathbb{R}$ with the real constant $c=\sqrt{2}$ and the companion $f=C_{j}$. By the same clause of Cell Integrals of the Trigonometric Monomials $|C_{j}(t)|\le1$, so by claim 4 of Properties of the Absolute Value in an Ordered Field, Claim 1 and claim 5 of Elementary Arithmetic in an Ordered Field, applied with the nonnegative factor $\sqrt{2}$,

For the periodicity, let $p\in\mathbb{Z}$. Then $jp\in\mathbb{Z}$ by claim 2 of Arithmetic, Order and Discreteness of the Integers, and by distributivity $2\pi j(t+p)=2\pi jt+2\pi(jp)$, so Quarter-Turn Identities and Periodicity of Sine and Cosine §integer, applied at the real number $2\pi jt$ with the integer $jp$, gives $C_{j}(t+p)=\cos(2\pi jt+2\pi(jp))=\cos(2\pi jt)=C_{j}(t)$, whence $\phi_{j}(t+p)=\phi_{j}(t)$.

Suppose $j<0$, so $-j\in\mathbb{Z}$ by claim 2 of Arithmetic, Order and Discreteness of the Integers and $\phi_{j}=\sqrt{2}\,S_{-j}$. The three assertions follow exactly as in the previous case, with $S_{-j}$ in place of $C_{j}$ and with the sine half of Cell Integrals of the Trigonometric Monomials §calculus and of Quarter-Turn Identities and Periodicity of Sine and Cosine §integer. This proves Claim 2.

Proof of claim 4 of the statement. Let $m\in\mathbb{Z}$. The bound $|\phi_{m}(t)|\le\sqrt{2}$ is Claim 2. For the derivative we use Cell Integrals of the Trigonometric Monomials §calculus, which gives $C_{j}'(t)=-2\pi j\,S_{j}(t)$ and $S_{j}'(t)=2\pi j\,C_{j}(t)$ for every $j\in\mathbb{Z}$, and claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, applied with the constant $\sqrt{2}$ and the relevant companion map.

If $m=0$ then $\phi_{0}$ is constant, so $\phi_{0}'(t)=0$ by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives; and $-2\pi\cdot0\cdot\phi_{0}(t)=0$ by claim 1 of Zero Products and Elementary Identities in a Field, so the identity holds.

If $0<m$ then $-m<0$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, so $\phi_{-m}=\sqrt{2}\,S_{-(-m)}=\sqrt{2}\,S_{m}$, and

the third equality by commutativity and associativity together with claim 2 of Zero Products and Elementary Identities in a Field.

If $m<0$ then $0<-m$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, so $\phi_{m}=\sqrt{2}\,S_{-m}$ and $\phi_{-m}=\sqrt{2}\,C_{-m}$, and

using claim 2 of Zero Products and Elementary Identities in a Field for $2\pi(-m)=-(2\pi m)$.

Proof of claim 2 of the statement. Let $a,b\in\mathbb{Z}$. By Claim 2 both $\phi_{a}$ and $\phi_{b}$ are continuous on $\mathbb{R}$, so $\phi_{a}\phi_{b}$ is continuous on $\mathbb{R}$ by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, applied on $\mathbb{R}$ to the two companions $\phi_{a}$ and $\phi_{b}$; hence its restriction to $J$ is $\mathcal{B}_{J}$-measurable and $\lambda_{J}$-integrable by Zero Extension of a Real-Valued Function, and the Unit-Cell Integral of a Continuous Function §continuous. The same applies to each of the maps $C_{a}$, $S_{a}$, $C_{a}C_{b}$, $S_{a}S_{b}$ and $S_{a}C_{b}$ appearing below, by Cell Integrals of the Trigonometric Monomials §calculus. Throughout we use claim 2 of Linearity and Monotonicity of the Lebesgue Integral, applied with a real constant and with the relevant integrable companion in both slots, to pull a constant factor out of an integral over $J$.

By claim 1 of Arithmetic, Order and Discreteness of the Integers each of $a$ and $b$ is zero, positive or negative, and we treat the nine resulting cases.

If $a=0$ and $b=0$ then $\phi_{a}\phi_{b}$ is the constant map $1$, which is $C_{0}C_{0}$ because $C_{0}$ is the constant map $1$ by Cell Integrals of the Trigonometric Monomials §calculus; so $\int_{J}\phi_{a}\phi_{b}\,d\lambda_{J}=1$ by Cell Integrals of the Trigonometric Monomials §cosine-cosine with $k=m=0$. Here $a=b$.

If $a=0$ and $0<b$ then $\phi_{a}\phi_{b}=\sqrt{2}\,C_{b}$, so the integral is $\sqrt{2}\int_{J}C_{b}\,d\lambda_{J}=\sqrt{2}\cdot0=0$ by Cell Integrals of the Trigonometric Monomials §single and claim 1 of Zero Products and Elementary Identities in a Field; and $a\ne b$. If $a=0$ and $b<0$ then $\phi_{a}\phi_{b}=\sqrt{2}\,S_{-b}$ and the integral is $0$ for the same reason. The two cases with $b=0$ and $a\ne0$ are the same computations with the roles of $a$ and $b$ exchanged, the product being unchanged by commutativity.

If $0<a$ and $0<b$ then $\phi_{a}\phi_{b}=(\sqrt{2}\,\sqrt{2})\,C_{a}C_{b}=2\,C_{a}C_{b}$ by Claim 1, so the integral is $2\int_{J}C_{a}C_{b}\,d\lambda_{J}$. If $a=b$ then $a\ne0$ and $|a|=|b|$, so Cell Integrals of the Trigonometric Monomials §cosine-cosine gives $\int_{J}C_{a}C_{b}\,d\lambda_{J}=2^{-1}$ and the integral is $2\cdot2^{-1}=1$. If $a\ne b$ then $|a|=a\ne b=|b|$ by Absolute Value in an Ordered Field, since $a$ and $b$ are positive; so that clause gives $\int_{J}C_{a}C_{b}\,d\lambda_{J}=0$ and the integral is $0$ by claim 1 of Zero Products and Elementary Identities in a Field.

If $a<0$ and $b<0$ then $-a$ and $-b$ are positive by claim 4 of Elementary Order Arithmetic in an Ordered Field and lie in $\mathbb{Z}$ by claim 2 of Arithmetic, Order and Discreteness of the Integers, and $\phi_{a}\phi_{b}=2\,S_{-a}S_{-b}$, so the integral is $2\int_{J}S_{-a}S_{-b}\,d\lambda_{J}$. If $a=b$ then $-a=-b$ and $-a\ne0$, so Cell Integrals of the Trigonometric Monomials §sine-sine gives $2^{-1}$ and the integral is $1$. If $a\ne b$ then $-a\ne-b$; moreover $-a\ne-(-b)=b$, because $-a$ is positive and $b$ is negative; and $|-a|=-a\ne-b=|-b|$ by Absolute Value in an Ordered Field, since $-a$ and $-b$ are positive. So the remaining case of Cell Integrals of the Trigonometric Monomials §sine-sine, taken with $k=-a$ and $m=-b$, gives $\int_{J}S_{-a}S_{-b}\,d\lambda_{J}=0$, and the integral is $0$.

If $0<a$ and $b<0$ then $\phi_{a}\phi_{b}=2\,C_{a}S_{-b}=2\,S_{-b}C_{a}$, so the integral is $2\int_{J}S_{-b}C_{a}\,d\lambda_{J}=0$ by Cell Integrals of the Trigonometric Monomials §sine-cosine and claim 1 of Zero Products and Elementary Identities in a Field; and $a\ne b$, since $a$ is positive and $b$ is negative. The case $a<0$, $0<b$ is the same with the roles exchanged. This exhausts the nine cases and proves claim 2 of the statement.

Claim 3. (Coordinate projections are continuous.) Let $i\in[n]$ and let $\pi_{i}:\mathbb{R}^{n}\to\mathbb{R}$ be the map $\pi_{i}(x)=x_{i}$. Then $\pi_{i}$ is continuous on $\mathbb{R}^{n}$ relative to $\mathbb{R}^{n}$, as a map from $(\mathbb{R}^{n},d_{E})$ into $\mathbb{R}$ with the absolute value metric $d_{\mathbb{R}}$.

Proof of Claim 3. Let $x\in\mathbb{R}^{n}$ and let $\varepsilon$ be a positive real number; choose $\delta=\varepsilon$. Let $y\in\mathbb{R}^{n}$ satisfy $d_{E}(x,y)<\delta$. By claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $d_{E}(x,y)=\lVert x-y\rVert$, and the $i$th coordinate of $x-y$ is $x_{i}-y_{i}$ by the definition of the difference of points; so claim 4 of that lemma gives

This is the condition of Continuous Map Between Metric Spaces at $x$ relative to $\mathbb{R}^{n}$, and $x$ was arbitrary. This proves Claim 3.

Proof of claim 1 of the statement. Let $k\in\mathbb{Z}^{n}$, so that $k_{i}\in\mathbb{Z}$ for every $i\in[n]$ by Lattice-Periodic Functions and the Periodic Function Classes §lattice.

First, $e_{k}$ is continuous on $\mathbb{R}^{n}$, in the sense fixed in Lattice-Periodic Functions and the Periodic Function Classes. Indeed, for $i\in[n]$ the map $\phi_{k_{i}}\circ\pi_{i}$, whose value at $x$ is $\phi_{k_{i}}(x_{i})$, is continuous on $\mathbb{R}^{n}$ by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, applied with the metric spaces $(\mathbb{R}^{n},d_{E})$ and $(\mathbb{R},d_{\mathbb{R}})$, with $g=\pi_{i}$ continuous by Claim 3, and with $u=\phi_{k_{i}}$ continuous by Claim 2. Now let $p_{j}:\mathbb{R}^{n}\to\mathbb{R}$ be $p_{j}(x)=\prod_{i=1}^{j}\phi_{k_{i}}(x_{i})$ for $j\in[n]$. By claim 1 of Properties of Finite Products, $p_{1}=\phi_{k_{1}}\circ\pi_{1}$ and $p_{S(r)}$ is the pointwise product of $p_{r}$ and $\phi_{k_{S(r)}}\circ\pi_{S(r)}$ whenever $S(r)\in[n]$, where $S$ is the successor map of Natural Numbers. An induction on $j$ bounded by $n$ — applying Principle of Induction for the Natural Numbers to the set of natural numbers $j$ such that either $n<j$, or $j\le n$ and $p_{j}$ is continuous, which contains $1$ and is closed under the successor because $j<S(j)\le n$ forces $j\le n$ by claim 1 of Properties of the Order on the Natural Numbers — gives, using claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space at each successor step, that $p_{n}=e_{k}$ is continuous on $\mathbb{R}^{n}$.

Second, $e_{k}$ is $\mathbb{Z}^{n}$-periodic. Let $x\in\mathbb{R}^{n}$ and $m\in\mathbb{Z}^{n}$. The $i$th coordinate of $x+m$ is $x_{i}+m_{i}$ by the definition of the sum of points, and $m_{i}\in\mathbb{Z}$ by Lattice-Periodic Functions and the Periodic Function Classes §lattice; so $\phi_{k_{i}}(x_{i}+m_{i})=\phi_{k_{i}}(x_{i})$ for every $i\in[n]$ by Claim 2. The two maps $i\mapsto\phi_{k_{i}}((x+m)_{i})$ and $i\mapsto\phi_{k_{i}}(x_{i})$ on $[n]$ therefore coincide, so $e_{k}(x+m)=e_{k}(x)$.

Hence $e_{k}\in C_{\mathrm{per}}$ by Lattice-Periodic Functions and the Periodic Function Classes §classes. By Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member, used with $p=2$, the restriction $e_{k}|_{Q}$ is measurable with respect to $\mathcal{B}_{Q}$ and belongs to $\mathcal{L}^{2}(\mathbb{T}^{n})$; the bound required there exists by Elementary Properties of Lattice-Periodic Functions §bounded, as that claim itself records. So $E_{k}=[\,e_{k}|_{Q}\,]$ is a well-defined member of $L^{2}(\mathbb{T}^{n})$.

Proof of claim 3 of the statement. Let $k,m\in\mathbb{Z}^{n}$. By The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, applied to the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ of The Flat Torus: Standing Notation §measure and to the members $e_{k}|_{Q}$ and $e_{m}|_{Q}$ of $\mathcal{L}^{2}(\mathbb{T}^{n})$ furnished by claim 1 of the statement,

the pointwise product being integrable by that clause. The maps $(e_{k}|_{Q})(e_{m}|_{Q})$ and $(e_{k}e_{m})|_{Q}$ agree, both taking the value $e_{k}(x)e_{m}(x)$ at $x\in Q$, by Restriction of a Map to a Subset.

For $i\in[n]$ let $g_{i}:J\to\mathbb{R}$ be the restriction to $J$ of $\phi_{k_{i}}\phi_{m_{i}}$. That map is continuous on $\mathbb{R}$ by claim 2 of the statement, so $g_{i}$ is measurable with respect to $\mathcal{B}_{J}$ by Zero Extension of a Real-Valued Function, and the Unit-Cell Integral of a Continuous Function §continuous; and by Claim 2 and claim 4 of Properties of the Absolute Value in an Ordered Field, together with Claim 1 and claim 5 of Elementary Arithmetic in an Ordered Field applied twice,

So the hypotheses of The Integral over the Unit Cell of a Product of One-Variable Functions are met with these $g_{i}$ and with the bounds $M_{i}=2$. For $x\in Q$ we have $x_{i}\in J$ for every $i\in[n]$, as recorded in The Integral over the Unit Cell of a Product of One-Variable Functions, and by claim 2 of Properties of Finite Products, applied to the two maps $i\mapsto\phi_{k_{i}}(x_{i})$ and $i\mapsto\phi_{m_{i}}(x_{i})$ on $[n]$,

so the map $G$ of that lemma is $(e_{k}e_{m})|_{Q}$. The clause The Integral over the Unit Cell of a Product of One-Variable Functions §integral therefore gives

By claim 2 of the statement the $i$th factor equals $1$ if $k_{i}=m_{i}$ and equals $0$ if $k_{i}\ne m_{i}$.

Points of $\mathbb{R}^{n}$ are $n$-tuples of real numbers, so $k=m$ if and only if $k_{i}=m_{i}$ for every $i\in[n]$. If $k=m$ then every factor equals $1$, so the product equals $1$ by claim 3 of Properties of Finite Products, applied with $i=1$. If $k\ne m$ then $k_{i}\ne m_{i}$ for some $i\in[n]$, so that factor is $0$ and the product is $0$ by claim 4 of Properties of Finite Products. This proves claim 3 of the statement.