Proof 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. The exponent of Continuous Periodic Functions are Power-Integrable and Dense on the Torus is taken to be $p=2$ throughout, and $\iota$ denotes the canonical map from $\mathbb{N}$ to $\mathbb{R}$.

An enumeration of the index set. The set $[2N+1]^{n}$ is nonempty and finite by claim 3 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets, as recorded in The Reproducing Identity for the Fejer Kernels of the Torus; being nonempty and finite, it has $M$ elements for some $M\in\mathbb{N}$, unique by Uniqueness of the Number of Elements, and there is a bijection $\varphi:[M]\to[2N+1]^{n}$. Fix one. By Sum over a Finite Index Set, for every map $f:[2N+1]^{n}\to\mathbb{R}$,

For $j\in[M]$ write $k_{j}=k(\varphi(j))\in\mathbb{Z}^{n}$, $W_{j}=W_{N,\varphi(j)}$ and $\hat{u}_{j}=\langle[\,u|_{Q}\,],E_{k_{j}}\rangle_{L^{2}}$. Applying the display to the map $a\mapsto W_{N,a}\langle[\,u|_{Q}\,],E_{k(a)}\rangle_{L^{2}}e_{k(a)}(x)$ for each fixed $x$ gives

and each $e_{k_{j}}$ lies in $C_{\mathrm{per}}$ by The Trigonometric System on the Torus is Orthonormal §classes.

Claim 1. By the last display, $\sigma_{N}u$ is the map of Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus formed with the coefficients $c_{j}=W_{j}\hat{u}_{j}$ and the maps $g_{j}=e_{k_{j}}$, so $\sigma_{N}u\in C_{\mathrm{per}}$ by Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus §member.

Claim 2. Let $U\in L^{2}(\mathbb{T}^{n})$. With the same data, Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus §pairing gives

since $[\,e_{k_{j}}|_{Q}\,]=E_{k_{j}}$. By the first display, applied to the map $a\mapsto W_{N,a}\langle[\,u|_{Q}\,],E_{k(a)}\rangle_{L^{2}}\langle U,E_{k(a)}\rangle_{L^{2}}$, the right-hand side is the sum over $[2N+1]^{n}$ asserted in claim 2.

Claim 3. Fix $x\in\mathbb{R}^{n}$ and let $y\in\mathbb{R}^{n}$. By The Reproducing Identity for the Fejer Kernels of the Torus §product, applied to the pair of points $y,x$, and the first display,

Multiplying by the real number $\mathbf{1}_{Q}(y)u(y)$ and moving it inside the sum by the homogeneity of finite sums, claim 3 of Properties of Finite Sums, then rearranging each summand by the commutativity and associativity of multiplication, gives

where $h_{j}:\mathbb{R}^{n}\to\mathbb{R}$ is the pointwise product. For each $j\in[M]$ the map $e_{k_{j}}u$ lies in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §algebra, so by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §integral the map $h_{j}=\mathbf{1}_{Q}(e_{k_{j}}u)$ is measurable and integrable, and

The restriction $(e_{k_{j}}u)|_{Q}$ is the pointwise product of $e_{k_{j}}|_{Q}$ and $u|_{Q}$, both of which lie in $\mathcal{L}^{2}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member; so by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product the right-hand side equals $\langle E_{k_{j}},[\,u|_{Q}\,]\rangle_{L^{2}}$, which is $\hat{u}_{j}$ by the symmetry of the inner product, condition (a) of Real Inner Product Space §inner-product. Now Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions, applied to the integrable maps $h_{j}$ and the coefficients $\beta_{j}$, shows by its clause Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable that $y\mapsto\mathbf{1}_{Q}(y)\Phi_{N}(y-x)u(y)$ is measurable and integrable, and by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear that

the middle equality by commutativity of multiplication and the last by the second display.

Claim 4. Let $0$ denote the origin of $\mathbb{R}^{n}$, all of whose coordinates are $0$; it lies in $\mathbb{Z}^{n}$ by Lattice-Periodic Functions and the Periodic Function Classes §lattice, since $0\in\mathbb{Z}$ by claim 2 of Arithmetic, Order and Discreteness of the Integers. By The Trigonometric System on the Torus §one-dimensional and The Trigonometric System on the Torus §system, $e_{0}(y)=\prod_{i=1}^{n}\phi_{0}(y_{i})=\prod_{i=1}^{n}1=1$ for every $y\in\mathbb{R}^{n}$, the last step by claim 3 of Properties of Finite Products applied with $i=1$ there, every factor other than the first being $1$. Also $e_{0}\in C_{\mathrm{per}}$ by The Trigonometric System on the Torus is Orthonormal §classes.

Fix $x\in\mathbb{R}^{n}$ and apply claim 3 to $u=e_{0}$: the map $y\mapsto\mathbf{1}_{Q}(y)\Phi_{N}(y-x)e_{0}(y)$, which is $y\mapsto\mathbf{1}_{Q}(y)\Phi_{N}(y-x)$ because $e_{0}(y)=1$, is measurable and integrable, and its integral equals $\sigma_{N}e_{0}(x)$. It remains to show $\sigma_{N}e_{0}(x)=1$.

Let $a^{0}\in[2N+1]^{n}$ be the tuple with $a^{0}_{i}=N+1$ for every $i\in[n]$; here $N+1\in[2N+1]$ because $N+1<(N+1)+N=N+(N+1)=2N+1$ by claim 6 of Properties of the Order on the Natural Numbers and claim 4 of Arithmetic of Addition on the Natural Numbers, hence $N+1\le2N+1$ by claim 1 of the former. The summands of $\sigma_{N}e_{0}(x)$ involve $\langle[\,e_{0}|_{Q}\,],E_{k(a)}\rangle_{L^{2}}=\langle E_{0},E_{k(a)}\rangle_{L^{2}}$, since $E_{0}=[\,e_{0}|_{Q}\,]$ by The Trigonometric System on the Torus is Orthonormal §classes. For $a\in[2N+1]^{n}$, by The Trigonometric System on the Torus is Orthonormal §orthonormal the number $\langle E_{0},E_{k(a)}\rangle_{L^{2}}$ equals $1$ if $k(a)=0$ and $0$ otherwise. Now $k(a)=0$ holds exactly when $k(a)_{i}=0$, that is $\iota(a_{i})=\iota(N)+1=\iota(N+1)$, for every $i\in[n]$, by claim 1 of Euclidean Points as Tuples of Real Numbers and claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; by the injectivity of $\iota$, claim 7 of that lemma, this holds exactly when $a_{i}=N+1$ for every $i\in[n]$, that is, exactly when $a=a^{0}$, two maps on $[n]$ being equal when their values agree everywhere. Hence the summand $W_{N,a}\langle E_{0},E_{k(a)}\rangle_{L^{2}}e_{k(a)}(x)$ of $\sigma_{N}e_{0}(x)$ vanishes for every $a\ne a^{0}$, and claims 4 and 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, applied with $E=\{a^{0}\}$, give

Finally, the integer attached to $a^{0}_{i}=N+1$ in The Reproducing Identity for the Fejer Kernels of the Torus is $\iota(N+1)-\iota(N)-1=0$, so $w_{N,N+1}=1-\tfrac{|0|}{N}=1$ by Absolute Value in an Ordered Field, and $W_{N,a^{0}}=\prod_{i=1}^{n}1=1$ by claim 3 of Properties of Finite Products as above. This proves claim 4.