Proof of Uniform Convergence of the Fejer Means of a Continuous Periodic Function on the Torus

Each result cited is universally quantified over the data in its own statement. We use the notation of The Flat Torus: Standing Notation: the initial segments $[n]$, points of $\mathbb{R}^{n}$ read as maps on $[n]$, the sum and difference of points, formed coordinatewise, the Euclidean norm $\lVert\,\cdot\,\rVert$, the lattice $\mathbb{Z}^{n}$, the cell $Q=\{y\in\mathbb{R}^{n}:0\le y_{i}<1\text{ for every }i\in[n]\}$ of The Half-Open Unit Cell Tiles Euclidean Space (i), which lies in $\mathcal{B}(\mathbb{R}^{n})$ by that clause, the wrapping map $\pi$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ of The Flat Torus: Standing Notation (iii), which is the restriction of $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$ to $Q$ furnished by claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, with the integral $\int_{\mathbb{T}^{n}}v\,dx$, and Lebesgue measure $\lambda_{n}$ on $\mathcal{B}(\mathbb{R}^{n})$. Natural numbers are read in $\mathbb{R}$ through the canonical map $\iota$, and the quotient $b/N$ of a real number $b$ by $N\in\mathbb{N}$ is the one fixed in The Real Numbers: Standing Notation and Background (i), namely $b\,\iota(N)^{-1}$, the inverse existing by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. The symbol $\sigma_{N}u$ is the Fejer mean, not the constant $\sigma_{n}$ of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (iii). Measurability and integrability of maps on $\mathbb{R}^{n}$, and integrals $\int_{\mathbb{R}^{n}}$, refer to $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$ as in The Fejer Means of a Continuous Periodic Function on the Torus; integrability includes measurability. $\mathbf{1}_{A}$ is the indicator of $A$, and $|t|$ the absolute value of $t\in\mathbb{R}$. Since the sum and difference of points are formed coordinatewise and two points with the same coordinates are equal, by claim 1 of Euclidean Points as Tuples of Real Numbers, identities such as $(y-m)+m=y$ and $(y-m)-x=(y-x)-m$ hold in $\mathbb{R}^{n}$ because they hold coordinatewise in the field $\mathbb{R}$. Let $\Phi_{N}$ be the Fejer kernel of the torus of order $N$. Transitivity of $\le$ on $\mathbb{R}$ is part of its being a total order, and the weak addition of inequalities $a\le b$, $c\le d\Rightarrow a+c\le b+d$ follows from claims 2 and 3 of Elementary Arithmetic in an Ordered Field; both are used without further comment.

Fix $u\in C_{\mathrm{per}}$. By Elementary Properties of Lattice-Periodic Functions (iii) there is a real number $M$ with $0\le M$ and $|u(z)|\le M$ for every $z\in\mathbb{R}^{n}$.

The explicit statement. Let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$, and put $\theta=\varepsilon\cdot2^{-1}$, so that $0<\theta$ and $\theta+\theta=\varepsilon$ by claim 8 of Elementary Order Arithmetic in an Ordered Field.

Step 1: uniform continuity. By A Continuous Lattice-Periodic Function is Uniformly Continuous (i), applied to $u$ and $\theta$, there is a real number $r$ with $0<r$ such that $|u(z)-u(z')|\le\theta$ for all $z,z'\in\mathbb{R}^{n}$ with $\lVert z-z'\rVert\le r$.

Step 2: the choice of $\delta$. Put $s_{n}=\sum_{i=1}^{n}1$, a finite sum of real numbers. Since $0\le1$ by claim 1 of Elementary Arithmetic in an Ordered Field, claim 6 of Properties of Finite Sums gives $1\le s_{n}$, hence $0<s_{n}$ by claim 6 and claim 2 of Elementary Order Arithmetic in an Ordered Field; so $s_{n}^{-1}$ exists and is positive by claim 7 of that lemma, and $0<r\,s_{n}^{-1}$ by claim 5. Put $\delta_{1}=2^{-1}\cdot2^{-1}$, which is positive by claims 8 and 5 of Elementary Order Arithmetic in an Ordered Field and satisfies $2\delta_{1}=2^{-1}<1$, the inequality being claim 8 applied with $\varepsilon=1$ there. By claim 9 of Elementary Order Arithmetic in an Ordered Field let $\delta$ be a least element of $\delta_{1}$ and $r\,s_{n}^{-1}$: $\delta\le\delta_{1}$, $\delta\le r\,s_{n}^{-1}$, and $\delta$ equals one of the two, so $0<\delta$. Then $2\delta\le2\delta_{1}$ by claim 5 of Elementary Arithmetic in an Ordered Field, so $2\delta<1$ by claim 2 of Elementary Order Arithmetic in an Ordered Field; and $\delta\,s_{n}\le r\,s_{n}^{-1}s_{n}=r$ by claim 5 of Elementary Arithmetic in an Ordered Field. Consequently also $\delta\le1-\delta$, since $\delta+\delta=2\delta<1$ gives $\delta<1-\delta$ by claim 1 of Elementary Order Arithmetic in an Ordered Field.

Step 3: a norm bound. Let $z\in\mathbb{R}^{n}$ satisfy $|z_{i}|\le\delta$ for every $i\in[n]$. Then $\lVert z\rVert\le r$. Indeed, for every $i$, $z_{i}^{2}=|z_{i}|^{2}$ by claim 1 of Nonnegativity of Squares in an Ordered Field, and $|z_{i}|^{2}\le\delta^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, since $0\le|z_{i}|$ by claim 1 of Properties of the Absolute Value in an Ordered Field. Hence each difference $\delta^{2}-z_{i}^{2}$ is nonnegative by claim 3 of Elementary Arithmetic in an Ordered Field, their sum is nonnegative by claim 5 of Properties of Finite Sums, and by claims 2 and 3 of that lemma and claim 3 of Elementary Arithmetic in an Ordered Field,

$$\lVert z\rVert^{2}=\sum_{i=1}^{n}z_{i}^{2}\le\sum_{i=1}^{n}\delta^{2}=\delta^{2}s_{n},$$

the first equality 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. Since $1\le s_{n}$ and $0\le\delta^{2}s_{n}$ by claim 2 of Nonnegativity of Squares in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field, the latter claim gives $\delta^{2}s_{n}\le\delta^{2}s_{n}s_{n}=(\delta s_{n})^{2}$, and $(\delta s_{n})^{2}\le r^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, since $0\le\delta s_{n}\le r$. Thus $\lVert z\rVert^{2}\le r^{2}$, and since $0\le\lVert z\rVert$ 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 and $0\le r$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert z\rVert\le r$.

Step 4: the far-field data. Apply The Fejer Kernel of the Torus: Nonnegativity, Periodicity, Mass and the Far-Field Integral Bound (v) with this $\delta$: with $K_{\delta}$ the closed interval determined by $\delta$ and $1-\delta$, that is, the set of $t\in\mathbb{R}$ with $\delta\le t\le1-\delta$, and $E_{\delta}=\{y\in Q:y_{i}\in K_{\delta}\text{ for some }i\in[n]\}$, one has $E_{\delta}\in\mathcal{B}_{Q}$, for every $N\in\mathbb{N}$ the map $g_{N}=\mathbf{1}_{E_{\delta}}\Phi_{N}|_{Q}$ on $Q$ is integrable with respect to $\lambda_{Q}$, and there is a real number $b_{\delta}$ with $0\le b_{\delta}$ and $\int_{\mathbb{T}^{n}}g_{N}\,dx\le b_{\delta}/N$ for every $N\in\mathbb{N}$. Each $g_{N}$ is nonnegative, as the product of the indicator and the kernel, which is nonnegative by The Fejer Kernel of the Torus: Nonnegativity, Periodicity, Mass and the Far-Field Integral Bound (ii), by claim 5 of Elementary Arithmetic in an Ordered Field.

Step 5: the choice of $N_{0}$. By claim 1 of The Archimedean Property of the Real Numbers there is $N_{0}\in\mathbb{N}$ with $2Mb_{\delta}\theta^{-1}<\iota(N_{0})$, where $\theta^{-1}$ exists by claim 7 of Elementary Order Arithmetic in an Ordered Field. Let $N\in\mathbb{N}$ with $N_{0}\le N$. Then $\iota(N_{0})\le\iota(N)$, by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field if $N_{0}<N$ and trivially if $N_{0}=N$; so $2Mb_{\delta}\theta^{-1}<\iota(N)$ by claim 2 of Elementary Order Arithmetic in an Ordered Field. Multiplying by $\theta$ and then by $\iota(N)^{-1}$, both positive by claims 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field and 7 of Elementary Order Arithmetic in an Ordered Field, claim 10 of the latter gives

$$\frac{2Mb_{\delta}}{N}<\theta .$$

Step 6: the basic identity. Fix $N\in\mathbb{N}$ with $N_{0}\le N$ and $x\in\mathbb{R}^{n}$. Let $\rho:\mathbb{R}^{n}\to\mathbb{R}$ be $\rho(y)=\mathbf{1}_{Q}(y)\Phi_{N}(y-x)$. By The Fejer Means of a Continuous Periodic Function on the Torus (iv), $\rho$ is integrable with $\int_{\mathbb{R}^{n}}\rho\,d\lambda_{n}=1$, and by The Fejer Means of a Continuous Periodic Function on the Torus (iii) the pointwise product $\rho u$ is integrable with $\int_{\mathbb{R}^{n}}\rho u\,d\lambda_{n}=\sigma_{N}u(x)$. Moreover $0\le\rho(y)$ for every $y$, by The Fejer Kernel of the Torus: Nonnegativity, Periodicity, Mass and the Far-Field Integral Bound (ii), $0\le\mathbf{1}_{Q}(y)$ and claim 5 of Elementary Arithmetic in an Ordered Field. By claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, applied with $a=1$ and $b=-u(x)$,

$$\sigma_{N}u(x)-u(x)=\int_{\mathbb{R}^{n}}\rho u\,d\lambda_{n}-u(x)\int_{\mathbb{R}^{n}}\rho\,d\lambda_{n}=\int_{\mathbb{R}^{n}}\bigl(\rho u-u(x)\rho\bigr)\,d\lambda_{n},$$

and by the same claim $|\sigma_{N}u(x)-u(x)|\le\int_{\mathbb{R}^{n}}|\rho u-u(x)\rho|\,d\lambda_{n}$.

Let $\psi:\mathbb{R}^{n}\to\mathbb{R}$ be the map $\psi=|\mathbf{1}_{Q}u-u(x)\mathbf{1}_{Q}|$, formed pointwise. It is measurable: $\mathbf{1}_{Q}u$ is measurable by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (ii), $\mathbf{1}_{Q}$ is measurable by claim 1 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps since $Q\in\mathcal{B}(\mathbb{R}^{n})$, and claims 2 and 4 of that lemma apply. For $y\in Q$ one has $\psi(y)=|u(y)-u(x)|$, and for $y\notin Q$ one has $\psi(y)=0$. Consequently, for every $y\in\mathbb{R}^{n}$,

$$|\rho(y)u(y)-u(x)\rho(y)|=|\rho(y)|\,|u(y)-u(x)|=\rho(y)\,\psi(y),$$

by distributivity, claim 4 of Properties of the Absolute Value in an Ordered Field, $|\rho(y)|=\rho(y)$ from Absolute Value in an Ordered Field, and the fact that both sides vanish for $y\notin Q$. So $\rho\psi=|\rho u-u(x)\rho|$ is integrable: it is measurable by claim 4 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, and its absolute value is itself, whose integral is finite because $\rho u-u(x)\rho$ is integrable, so the criterion of Integrable Functions and Their Lebesgue Integral applies. Hence $|\sigma_{N}u(x)-u(x)|\le\int_{\mathbb{R}^{n}}\rho\psi\,d\lambda_{n}$. Finally $0\le\psi(y)\le2M$ for every $y$: nonnegativity is claim 1 of Properties of the Absolute Value in an Ordered Field, the bound is trivial off $Q$, and for $y\in Q$ claims 5 and 2 of that lemma give $|u(y)-u(x)|\le|u(y)|+|u(x)|\le M+M=2M$.

Step 7: the splitting. Let $\tilde{g}_{N}:\mathbb{R}^{n}\to\mathbb{R}$ be the periodic extension of $g_{N}$, so that $\tilde{g}_{N}(z)=g_{N}(\pi(z))$. By The Periodic Extension of a Function on the Unit Cell (i) it is measurable and $\mathbb{Z}^{n}$-periodic, and $\tilde{g}_{N}(y)=g_{N}(y)$ for $y\in Q$. Hence $\mathbf{1}_{Q}\tilde{g}_{N}$ is the map equal to $g_{N}$ on $Q$ and to $0$ outside $Q$, that is, the zero extension of $g_{N}$ in the sense of Zero Extension of a Real-Valued Function, and the Unit-Cell Integral of a Continuous Function (i); so by that clause, applied to the measure space $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$, its restriction $(Q,\mathcal{B}_{Q},\lambda_{Q})$ and the map $g_{N}$, which is integrable with respect to $\lambda_{Q}$ by Step 4 and hence measurable with respect to $\mathcal{B}_{Q}$, the map $\mathbf{1}_{Q}\tilde{g}_{N}$ is integrable with $\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\tilde{g}_{N}\,d\lambda_{n}=\int_{\mathbb{T}^{n}}g_{N}\,dx$. For $z\in\mathbb{R}^{n}$, by The Half-Open Unit Cell Tiles Euclidean Space (iii) one has $\pi(z)=z-m$ for some $m\in\mathbb{Z}^{n}$ and $\pi(z)\in Q$, so that $\Phi_{N}(z)=\Phi_{N}((z-m)+m)=\Phi_{N}(z-m)=\Phi_{N}(\pi(z))$ by The Fejer Kernel of the Torus: Nonnegativity, Periodicity, Mass and the Far-Field Integral Bound (iii), and hence

$$\tilde{g}_{N}(z)=\mathbf{1}_{E_{\delta}}(\pi(z))\,\Phi_{N}(z)\qquad(z\in\mathbb{R}^{n}),$$

where $\mathbf{1}_{E_{\delta}}(\pi(z))$ is $1$ if $\pi(z)\in E_{\delta}$ and $0$ otherwise. By The Cell Integral of a Translated Periodic Function (i), applied to $w=\tilde{g}_{N}$ and $x$, the map $\tau:\mathbb{R}^{n}\to\mathbb{R}$, $\tau(y)=\mathbf{1}_{Q}(y)\tilde{g}_{N}(y-x)$, is integrable and

$$\int_{\mathbb{R}^{n}}\tau\,d\lambda_{n}=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\tilde{g}_{N}\,d\lambda_{n}=\int_{\mathbb{T}^{n}}g_{N}\,dx\le\frac{b_{\delta}}{N}.$$

Also $0\le\tau(y)$ for every $y$, since $g_{N}$ is nonnegative. For $y\in\mathbb{R}^{n}$ write $\chi(y)=\mathbf{1}_{E_{\delta}}(\pi(y-x))\in\{0,1\}$, so that $\tau(y)=\chi(y)\rho(y)$.

Let $A=\tau\psi$ and $B=\rho\psi-A$, formed pointwise; both are measurable by claims 2 and 3 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps. For every $y$, $A(y)=\chi(y)\rho(y)\psi(y)$ with $0\le\chi(y)\le1$, so $0\le A(y)\le\rho(y)\psi(y)$ by claim 5 of Elementary Arithmetic in an Ordered Field, and hence $0\le B(y)$ by claim 3 of that lemma; also $B(y)\le\rho(y)\psi(y)$ since $0\le A(y)$. As $\rho\psi$ is integrable, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, applied to the nonnegative maps $A$, $B$ and $\rho\psi$ with their values read in $[0,\infty]$, the two readings of measurability agreeing by Measurable Functions and the Lebesgue Integral: Standing Notation (iii), bounds the integrals of $A=|A|$ and $B=|B|$ by that of $\rho\psi$, which is finite; so $A$ and $B$ are integrable by Integrable Functions and Their Lebesgue Integral, and $\rho\psi=A+B$ gives, by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion,

$$\int_{\mathbb{R}^{n}}\rho\psi\,d\lambda_{n}=\int_{\mathbb{R}^{n}}A\,d\lambda_{n}+\int_{\mathbb{R}^{n}}B\,d\lambda_{n}.$$

Step 8: the far part. For every $y$, $A(y)=\tau(y)\psi(y)\le\tau(y)\,2M=2M\tau(y)$ by claim 5 of Elementary Arithmetic in an Ordered Field, since $0\le\tau(y)$ and $\psi(y)\le2M$. The map $2M\tau$ is integrable by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, and the same claim gives

$$\int_{\mathbb{R}^{n}}A\,d\lambda_{n}\le\int_{\mathbb{R}^{n}}2M\tau\,d\lambda_{n}=2M\int_{\mathbb{R}^{n}}\tau\,d\lambda_{n}\le2M\,\frac{b_{\delta}}{N}<\theta,$$

using $0\le2M$ with claim 5 of Elementary Arithmetic in an Ordered Field for the second inequality and Step 5 for the last.

Step 9: the near part. We claim $B(y)\le\theta\rho(y)$ for every $y\in\mathbb{R}^{n}$. If $y\notin Q$ then $\rho(y)=0$ and $\psi(y)=0$, so $B(y)=0=\theta\rho(y)$. If $y\in Q$ and $\chi(y)=1$, then $A(y)=\rho(y)\psi(y)$, so $B(y)=0\le\theta\rho(y)$ by claim 5 of Elementary Arithmetic in an Ordered Field. Suppose finally $y\in Q$ and $\chi(y)=0$, so that $A(y)=0$ and $B(y)=\rho(y)|u(y)-u(x)|$. We show $|u(y)-u(x)|\le\theta$, which gives $B(y)\le\rho(y)\theta=\theta\rho(y)$ by claim 5 of Elementary Arithmetic in an Ordered Field.

Put $w=y-x$ and $q=\pi(w)$; by The Half-Open Unit Cell Tiles Euclidean Space (iii), $q\in Q$ and $q=w-m$ for some $m\in\mathbb{Z}^{n}$. Since $\chi(y)=0$, $q\notin E_{\delta}$, so for every $i\in[n]$ the coordinate $q_{i}$ does not lie in $K_{\delta}$: it is not the case that both $\delta\le q_{i}$ and $q_{i}\le1-\delta$. Also $0\le q_{i}<1$ since $q\in Q$. The order of $\mathbb{R}$ being total, either $q_{i}<\delta$, or $\delta\le q_{i}$ and then $1-\delta<q_{i}$. Let $m'$ be the point of $\mathbb{R}^{n}$ with $m'_{i}=m_{i}$ if $q_{i}<\delta$ and $m'_{i}=m_{i}+1$ otherwise, which exists by claim 2 of Euclidean Points as Tuples of Real Numbers; every $m'_{i}$ is an integer by claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers, so $m'\in\mathbb{Z}^{n}$ by Lattice-Periodic Functions and the Periodic Function Classes (i). Put $w'=w-m'$, so that $w'_{i}=w_{i}-m'_{i}$ for every $i$. If $q_{i}<\delta$, then $w'_{i}=w_{i}-m_{i}=q_{i}$, and $0\le q_{i}$ gives $|w'_{i}|=q_{i}<\delta$ by Absolute Value in an Ordered Field. Otherwise $w'_{i}=w_{i}-m_{i}-1=q_{i}-1$, which is negative because $q_{i}<1$ (claim 1 of Elementary Order Arithmetic in an Ordered Field), so $|w'_{i}|=-(q_{i}-1)=1-q_{i}$ by Absolute Value in an Ordered Field, and $1-\delta<q_{i}$ gives $1-q_{i}<\delta$ by claim 1 of Elementary Order Arithmetic in an Ordered Field. In either case $|w'_{i}|\le\delta$, so $\lVert w'\rVert\le r$ by Step 3.

Now $(y-m')-x=w-m'=w'$, so $\lVert(y-m')-x\rVert\le r$ and Step 1 gives $|u(y-m')-u(x)|\le\theta$. Since $m'\in\mathbb{Z}^{n}$ and $u$ is $\mathbb{Z}^{n}$-periodic by Lattice-Periodic Functions and the Periodic Function Classes (iii) and Lattice-Periodic Functions and the Periodic Function Classes (ii), $u(y)=u((y-m')+m')=u(y-m')$. Hence $|u(y)-u(x)|\le\theta$, proving the claim.

The map $\theta\rho$ is integrable with integral $\theta$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and Step 6, and the same claim gives $\int_{\mathbb{R}^{n}}B\,d\lambda_{n}\le\int_{\mathbb{R}^{n}}\theta\rho\,d\lambda_{n}=\theta$.

Step 10: conclusion. By Steps 6 to 9 and claim 3 of Elementary Order Arithmetic in an Ordered Field,

$$|\sigma_{N}u(x)-u(x)|\le\int_{\mathbb{R}^{n}}A\,d\lambda_{n}+\int_{\mathbb{R}^{n}}B\,d\lambda_{n}<\theta+\theta=\varepsilon .$$

As $N\in\mathbb{N}$ with $N_{0}\le N$ and $x\in\mathbb{R}^{n}$ were arbitrary, this proves the explicit statement, with the inequality even strict.

Uniform convergence. Let $\varepsilon'\in\mathbb{R}$ with $0<\varepsilon'$. By claim 8 of Elementary Order Arithmetic in an Ordered Field the number $\varepsilon=\varepsilon'\cdot2^{-1}$ is positive and satisfies $\varepsilon<\varepsilon'$. The explicit statement furnishes $N_{0}\in\mathbb{N}$ with $|\sigma_{N}u(x)-u(x)|\le\varepsilon<\varepsilon'$, hence $|\sigma_{N}u(x)-u(x)|<\varepsilon'$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, for every $N\in\mathbb{N}$ with $N_{0}\le N$ and every $x\in\mathbb{R}^{n}$. This is the condition of Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps (ii) for the sequence $(\sigma_{N}u)_{N\in\mathbb{N}}$ of maps from $D=\mathbb{R}^{n}$ to $\mathbb{R}$, the limit $f=u$, the subset $S=\mathbb{R}^{n}$ and $K=N_{0}$.