The Fejer Means of a Continuous Periodic Function on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the initial segments $[n]$, Euclidean space $\mathbb{R}^{n}$ with the difference $y-x$ of its points, Lebesgue measure $\lambda_{n}$ on $\mathcal{B}(\mathbb{R}^{n})$, the integer lattice $\mathbb{Z}^{n}$, the cell $Q$ with the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ and the notation $\int_{\mathbb{T}^{n}}v\,dx$, the class $\mathcal{L}^{2}(\mathbb{T}^{n})$ and the space $L^{2}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic class $C_{\mathrm{per}}$ and the restriction $u|_{Q}$ are the ones fixed there. Measurability of a map on $\mathbb{R}^{n}$ means measurability with respect to $\mathcal{B}(\mathbb{R}^{n})$, integrals over $\mathbb{R}^{n}$ are integrals with respect to $\lambda_{n}$ in the measure space $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$, and $\mathbf{1}_{Q}$ is the indicator of $Q$. Let $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ be the inner product of $L^{2}(\mathbb{T}^{n})$, let $\phi_{m}$ for $m\in\mathbb{Z}$ be the one-dimensional trigonometric maps, let $e_{k}$ for $k\in\mathbb{Z}^{n}$ be the trigonometric system and $E_{k}=[\,e_{k}|_{Q}\,]$ its classes, as introduced in The Trigonometric System on the Torus is Orthonormal (i). Let $N\in\mathbb{N}$; let $\Phi_{N}$ be the Fejer kernel of the torus of order $N$, and let $[2N+1]^{n}$, $k(a)\in\mathbb{Z}^{n}$ and $W_{N,a}$ for $a\in[2N+1]^{n}$ be as in The Reproducing Identity for the Fejer Kernels of the Torus, where $[2N+1]^{n}$ is recorded to be nonempty and finite; a sum indexed by it is that of Sum over a Finite Index Set.

Let $u\in C_{\mathrm{per}}$, so that $u|_{Q}\in\mathcal{L}^{2}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i). The Fejer mean of order $N$ of $u$ is the map $\sigma_{N}u:\mathbb{R}^{n}\to\mathbb{R}$ given by

$$\sigma_{N}u(x)=\sum_{a\in[2N+1]^{n}}W_{N,a}\,\bigl\langle[\,u|_{Q}\,],E_{k(a)}\bigr\rangle_{L^{2}}\,e_{k(a)}(x)\qquad(x\in\mathbb{R}^{n}).$$

Then the following hold.

1. (Membership)(i) $\sigma_{N}u\in C_{\mathrm{per}}$.

2. (Pairing)(ii) For every $U\in L^{2}(\mathbb{T}^{n})$,

$$\bigl\langle U,[\,(\sigma_{N}u)|_{Q}\,]\bigr\rangle_{L^{2}}=\sum_{a\in[2N+1]^{n}}W_{N,a}\,\bigl\langle[\,u|_{Q}\,],E_{k(a)}\bigr\rangle_{L^{2}}\,\bigl\langle U,E_{k(a)}\bigr\rangle_{L^{2}} .$$

3. (The integral representation)(iii) Let $x\in\mathbb{R}^{n}$. The map $y\mapsto\mathbf{1}_{Q}(y)\,\Phi_{N}(y-x)\,u(y)$ on $\mathbb{R}^{n}$ is measurable and integrable, and

$$\sigma_{N}u(x)=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}(y)\,\Phi_{N}(y-x)\,u(y)\,d\lambda_{n}(y).$$

4. (Mass)(iv) Let $x\in\mathbb{R}^{n}$. The map $y\mapsto\mathbf{1}_{Q}(y)\,\Phi_{N}(y-x)$ on $\mathbb{R}^{n}$ is measurable and integrable, and

$$\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}(y)\,\Phi_{N}(y-x)\,d\lambda_{n}(y)=1 .$$