Proof of The Fejer Kernel of the Torus: Nonnegativity, Periodicity, Mass and the Far-Field Integral Bound

Each result cited is universally quantified over the data in its own statement. Fix $N\in\mathbb{N}$.

Claim 1. By The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay (iii) one has $0\le F_{N}(t)$ for every $t\in\mathbb{R}$, so it remains to produce an upper bound. Put $c_{m}=\tfrac{2(N-m)}{N}$ for $m\in\mathbb{N}$. The definition of $F_{N}$ in The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay reads $F_{N}(t)=\tfrac{1}{N}\bigl(N+2\sum_{m=1}^{N}(N-m)\cos(2\pi mt)\bigr)$; distributing $\tfrac{1}{N}$ over the sum of the two terms, using $\tfrac{1}{N}N=1$, and applying the homogeneity of finite sums, claim 3 of Properties of Finite Sums, twice to move the factor $\tfrac{2}{N}$ inside, gives

$$F_{N}(t)=C_{0}(t)+\sum_{m=1}^{N}c_{m}C_{m}(t)\qquad(t\in\mathbb{R}),$$

where $C_{m}(t)=\cos(2\pi mt)$ and $C_{0}(t)=1$ for every $t$, by Cell Integrals of the Trigonometric Monomials (i). Now let $P$ be the set of those $p\in\mathbb{N}$ for which there is a real number $M_{p}$ with

$$\Bigl|\sum_{m=1}^{p}c_{m}C_{m}(t)\Bigr|\le M_{p}\qquad\text{for every }t\in\mathbb{R},$$

where $c_{m}$ is given by the same formula for every $m\in\mathbb{N}$. Each $C_{m}$ satisfies $|C_{m}(t)|\le1$ by Cell Integrals of the Trigonometric Monomials (i), so claim 4 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field give $|c_{m}C_{m}(t)|=|c_{m}||C_{m}(t)|\le|c_{m}|$. A sum with one summand equals that summand, by claim 1 of Properties of Finite Sums, so $M_{1}=|c_{1}|$ witnesses $1\in P$. If $p\in P$ with witness $M_{p}$, the recursion in claim 1 of Properties of Finite Sums and the triangle inequality, claim 5 of Properties of the Absolute Value in an Ordered Field, give

$$\Bigl|\sum_{m=1}^{p+1}c_{m}C_{m}(t)\Bigr|\le\Bigl|\sum_{m=1}^{p}c_{m}C_{m}(t)\Bigr|+|c_{p+1}C_{p+1}(t)|\le M_{p}+|c_{p+1}|,$$

so $M_{p+1}=M_{p}+|c_{p+1}|$ witnesses $p+1\in P$. By Principle of Induction for the Natural Numbers, $P=\mathbb{N}$. Since $C_{0}(t)=1$ for every $t$, by Cell Integrals of the Trigonometric Monomials (i), the number $M=1+M_{N}$ satisfies $F_{N}(t)\le M$ for every $t$, using claim 3 of Properties of the Absolute Value in an Ordered Field and the compatibility of the order with addition, claim 1 of Elementary Order Arithmetic in an Ordered Field. Since $0\le F_{N}(t)$, the definition of the absolute value gives $|F_{N}(t)|=F_{N}(t)\le M$ for every $t\in\mathbb{R}$.

Now $F_{N}|_{J}$ is measurable with respect to $\mathcal{B}_{J}$: it is $\lambda_{J}$-integrable by The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay (i), and measurability is part of integrability by Integrable Functions and Their Lebesgue Integral. Taking $g_{i}=F_{N}|_{J}$ for every $i\in[n]$ in The Integral over the Unit Cell of a Product of One-Variable Functions, the map $G$ of its claim 1 is exactly $\Phi_{N}|_{Q}$, because a point of $Q$ has all its coordinates in $J$; so $\Phi_{N}|_{Q}$ is measurable with respect to $\mathcal{B}_{Q}$ by The Integral over the Unit Cell of a Product of One-Variable Functions (i), and, the bound $|F_{N}(t)|\le M$ of the previous paragraph serving as the hypothesis $|g_{i}(t)|\le M_{i}$ of The Integral over the Unit Cell of a Product of One-Variable Functions (ii) with $M_{i}=M$, it is $\lambda_{Q}$-integrable by that clause.

Claim 2. Let $y\in\mathbb{R}^{n}$. Every factor $F_{N}(y_{i})$ is nonnegative by The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay (iii), so the product is nonnegative by claim 5 of Properties of Finite Products, which gives nonnegativity of a finite product of nonnegative real numbers.

Claim 3. Let $y\in\mathbb{R}^{n}$ and $k\in\mathbb{Z}^{n}$. For every $i\in[n]$ the component $k_{i}$ lies in $\mathbb{Z}$, by Lattice-Periodic Functions and the Periodic Function Classes (i), and $(y+k)_{i}=y_{i}+k_{i}$ because the sum of two points of $\mathbb{R}^{n}$ is formed coordinatewise; so $F_{N}((y+k)_{i})=F_{N}(y_{i})$ by The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay (i). The two products therefore have equal factors at every index, and are equal.

Claim 4. Taking $g_{i}=F_{N}|_{J}$ for every $i\in[n]$ in The Integral over the Unit Cell of a Product of One-Variable Functions (ii), whose hypotheses were checked in claim 1,

$$\int_{\mathbb{T}^{n}}\Phi_{N}\,dx=\prod_{i=1}^{n}\int_{J}F_{N}\,d\lambda_{J}.$$

Each factor equals $1$ by The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay (iv), so the product equals $1$ by claim 2 of Properties of Natural Number Powers in a Field, the product of $n$ copies of $1$ being $1^{n}$ by Natural Number Power of an Element of a Field.

Claim 5. Let $c_{\delta}$ be a positive real number furnished by The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay (v) for this $\delta$. The interval $K_{\delta}$ is a Borel subset of $\mathbb{R}$ by Borel Sigma-Algebra on the Real Line, being an interval, and $K_{\delta}\subseteq J$: indeed $0<\delta$ and $\delta\le t$ give $0\le t$ for $t\in K_{\delta}$, while $t\le1-\delta$ and $1-\delta<1$, the latter because $0<\delta$ and claim 1 of Elementary Order Arithmetic in an Ordered Field, give $t<1$ by claim 2 of that lemma. Hence $K_{\delta}\in\mathcal{B}_{J}$.

For $i\in[n]$ put $A_{i}=\{y\in Q:y_{i}\in K_{\delta}\}$, and let $h:J\to\mathbb{R}$ be $\mathbf{1}_{K_{\delta}}F_{N}|_{J}$, which is measurable with respect to $\mathcal{B}_{J}$ by claim 1 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, giving measurability of an indicator, and claim 3 of that lemma, giving measurability of a product, and which satisfies $|h(t)|\le M$ for every $t\in J$ with the $M$ of claim 1, by claim 4 of Properties of the Absolute Value in an Ordered Field and $0\le\mathbf{1}_{K_{\delta}}(t)\le1$. Fix $i\in[n]$ and take $g_{i}=h$ and $g_{j}=F_{N}|_{J}$ for $j\in[n]$ with $j\ne i$ in The Integral over the Unit Cell of a Product of One-Variable Functions. The map $G$ of its claim 1 is $\mathbf{1}_{A_{i}}\Phi_{N}|_{Q}$: a point $y\in Q$ lies in $A_{i}$ exactly when $y_{i}\in K_{\delta}$, in which case the two products agree factor by factor, while if $y\notin A_{i}$ then $g_{i}(y_{i})=0$ and $G(y)=0$ by claim 4 of Properties of Finite Products. So that map is measurable and $\lambda_{Q}$-integrable, and The Integral over the Unit Cell of a Product of One-Variable Functions (ii) gives

$$\int_{\mathbb{T}^{n}}\mathbf{1}_{A_{i}}\Phi_{N}\,dx=\prod_{j=1}^{n}\int_{J}g_{j}\,d\lambda_{J},$$

where every factor with $j\ne i$ equals $1$ by The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay (iv); so, by claim 3 of Properties of Finite Products, which evaluates a finite product all of whose factors but one equal $1$,

$$\int_{\mathbb{T}^{n}}\mathbf{1}_{A_{i}}\Phi_{N}\,dx=\int_{J}h\,d\lambda_{J}.$$

By The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay (v) one has $F_{N}(t)\le\tfrac{1}{N\,c_{\delta}}$ for $t\in K_{\delta}$, and $h$ vanishes off $K_{\delta}$ while $0\le h$ by The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay (iii) and claim 5 of Elementary Arithmetic in an Ordered Field; so $0\le h(t)\le\tfrac{1}{N\,c_{\delta}}$ for every $t\in J$, the number $\tfrac{1}{N\,c_{\delta}}$ being positive by claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field. Let $\ell:J\to\mathbb{R}$ be the constant map with value $\tfrac{1}{N\,c_{\delta}}$, so that $\ell=\tfrac{1}{N\,c_{\delta}}\mathbf{1}_{J}$; the indicator $\mathbf{1}_{J}$ is measurable by claim 1 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, equals its own absolute value, and has integral $\lambda_{J}(J)=1$ by The Integral of an Indicator Function is the Measure of the Set and $\lambda_{J}(J)=1$ as recorded in The Integral over the Unit Cell of a Product of One-Variable Functions, so it is integrable by Integrable Functions and Their Lebesgue Integral, and then $\ell$ is integrable with $\int_{J}\ell\,d\lambda_{J}=\tfrac{1}{N\,c_{\delta}}$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion. The map $h$ is integrable, being the factor $g_{i}$ in the application of The Integral over the Unit Cell of a Product of One-Variable Functions (ii) above, which asserts the integrability of each factor. Monotonicity of the integral for integrable maps, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, therefore gives

$$\int_{J}h\,d\lambda_{J}\le\int_{J}\ell\,d\lambda_{J}=\frac{1}{N\,c_{\delta}} .$$

Measurability of the far-field set. Let $\pi_{i}:\mathbb{R}^{n}\to\mathbb{R}$ be the $i$th coordinate map, which is measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$ by claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, the $\sigma$-algebra there being $\mathcal{B}(\mathbb{R}^{n})$ by claim 5 of that lemma, so that $\pi_{i}^{-1}(K_{\delta})\in\mathcal{B}(\mathbb{R}^{n})$. A point $y\in Q$ lies in $A_{i}$ exactly when $y_{i}\in K_{\delta}$, that is, $A_{i}=\pi_{i}^{-1}(K_{\delta})\cap Q$. The cell satisfies $Q\in\mathcal{B}(\mathbb{R}^{n})$, as fixed in The Flat Torus: Standing Notation (iii), and a $\sigma$-algebra is closed under finite intersections by σ-Algebra and Measurable Space; hence $A_{i}\in\mathcal{B}(\mathbb{R}^{n})$, and since $A_{i}\subseteq Q$ the description of $\mathcal{B}_{Q}$ as the restriction furnished by claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, adopted in The Flat Torus: Standing Notation (iii), gives $A_{i}\in\mathcal{B}_{Q}$. The set $E_{\delta}$ is the union of the $A_{i}$ over $i\in[n]$, so $E_{\delta}\in\mathcal{B}_{Q}$ by the same closure properties. Consequently $\mathbf{1}_{E_{\delta}}$ and each $\mathbf{1}_{A_{i}}$ are measurable with respect to $\mathcal{B}_{Q}$ by claim 1 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, and the products $\mathbf{1}_{E_{\delta}}\Phi_{N}|_{Q}$ and $\mathbf{1}_{A_{i}}\Phi_{N}|_{Q}$ are measurable by claim 3 of that lemma.

Integrability. For every $y\in Q$ one has $0\le\mathbf{1}_{E_{\delta}}(y)\le1$ and $0\le\Phi_{N}(y)$ by claim 2, so $0\le\mathbf{1}_{E_{\delta}}(y)\Phi_{N}(y)\le\Phi_{N}(y)$ by claim 5 of Elementary Arithmetic in an Ordered Field. The map $\Phi_{N}|_{Q}$ is integrable by claim 1, so monotonicity of the integral of nonnegative maps, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, applied with the nonnegative real values read in $[0,\infty]$, the two readings of measurability agreeing by Measurable Functions and the Lebesgue Integral: Standing Notation (iii), bounds the integral of the absolute value of $\mathbf{1}_{E_{\delta}}\Phi_{N}|_{Q}$, which is the map itself, by that of $\Phi_{N}|_{Q}$, which is finite; hence $\mathbf{1}_{E_{\delta}}\Phi_{N}|_{Q}$ is integrable by Integrable Functions and Their Lebesgue Integral.

The bound. Let $y\in Q$. If $y\in E_{\delta}$ then $y\in A_{j}$ for some $j\in[n]$, so $\mathbf{1}_{A_{j}}(y)=1$; the summands $\mathbf{1}_{A_{i}}(y)$ are all nonnegative, so claim 6 of Properties of Finite Sums, which bounds a nonnegative summand by the whole sum, gives $1\le\sum_{i=1}^{n}\mathbf{1}_{A_{i}}(y)$, and the left-hand side below is $1$. If $y\notin E_{\delta}$ the left-hand side below is $0$, while the sum is nonnegative by claim 5 of Properties of Finite Sums, which gives nonnegativity of a finite sum of nonnegative summands. In either case

$$\mathbf{1}_{E_{\delta}}(y)\le\sum_{i=1}^{n}\mathbf{1}_{A_{i}}(y).$$

Multiplying by the nonnegative $\Phi_{N}(y)$, by claim 5 of Elementary Arithmetic in an Ordered Field, and moving the factor inside the sum by the homogeneity of finite sums, claim 3 of Properties of Finite Sums, gives $\mathbf{1}_{E_{\delta}}(y)\Phi_{N}(y)\le\sum_{i=1}^{n}\mathbf{1}_{A_{i}}(y)\Phi_{N}(y)$ for every $y\in Q$. The summands $\mathbf{1}_{A_{i}}\Phi_{N}|_{Q}$ are integrable by claim 5, so the sum is integrable by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions (i); integrating, by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion for monotonicity of the integral of integrable maps and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions (ii) for the integral of the finite sum,

$$\int_{\mathbb{T}^{n}}\mathbf{1}_{E_{\delta}}\Phi_{N}\,dx\le\sum_{i=1}^{n}\int_{\mathbb{T}^{n}}\mathbf{1}_{A_{i}}\Phi_{N}\,dx .$$

Each summand on the right is at most $\tfrac{1}{N\,c_{\delta}}$, by the two displays already established. Writing $d_{i}$ for the difference between $\tfrac{1}{N\,c_{\delta}}$ and the $i$th summand, each $d_{i}$ is nonnegative by claim 3 of Elementary Arithmetic in an Ordered Field, so $0\le\sum_{i=1}^{n}d_{i}$ by claim 5 of Properties of Finite Sums, and additivity, claim 2 of that lemma, with claim 3 of Elementary Arithmetic in an Ordered Field again, turns this into

$$\sum_{i=1}^{n}\int_{\mathbb{T}^{n}}\mathbf{1}_{A_{i}}\Phi_{N}\,dx\le\sum_{i=1}^{n}\frac{1}{N\,c_{\delta}}=\frac{1}{N}\sum_{i=1}^{n}\frac{1}{c_{\delta}},$$

the last equality by claim 3 of Properties of Finite Sums. Put $b_{\delta}=\sum_{i=1}^{n}\tfrac{1}{c_{\delta}}$; each summand is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, so $b_{\delta}$ is a nonnegative real number by claim 5 of Properties of Finite Sums, and it depends only on $\delta$ and $n$. Combining the two displays gives the asserted bound.