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

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 $\mathbb{R}^{n}$, whose points are read as maps on $[n]$ with real values, the integer lattice $\mathbb{Z}^{n}$, the half-open unit cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ with the notation $\int_{\mathbb{T}^{n}}v\,dx$, and the integral and the notion of an integrable map are the ones fixed there. Let $J$ and $(J,\mathcal{B}_{J},\lambda_{J})$ be as in The Integral over the Unit Cell of a Product of One-Variable Functions, so that a point $x\in\mathbb{R}^{n}$ lies in $Q$ exactly when $x_{i}\in J$ for every $i\in[n]$. Finite products and finite sums are those of The Real Numbers: Standing Notation and Background (ii) and Finite Product Notation in a Field, the sum $y+k$ of two points of $\mathbb{R}^{n}$ is the coordinatewise sum, natural numbers are read in $\mathbb{R}$ through the canonical map fixed in The Real Numbers: Standing Notation and Background (i), so that $b/N$ denotes the quotient of a real number $b$ by the positive image of $N$, and $\mathbf{1}_{A}$ denotes the indicator function of a set $A$.

For $N\in\mathbb{N}$ let $F_{N}$ be the Fejer kernel of order $N$ fixed there, and define $\Phi_{N}:\mathbb{R}^{n}\to\mathbb{R}$, the Fejer kernel of the torus of order $N$, by

$$\Phi_{N}(y)=\prod_{i=1}^{n}F_{N}(y_{i})\qquad(y\in\mathbb{R}^{n}).$$

Then the following hold for every $N\in\mathbb{N}$.

1. (Measurability, boundedness and integrability)(i) There is a real number $M$ with $0\le F_{N}(t)\le M$ for every $t\in\mathbb{R}$. The restriction $\Phi_{N}|_{Q}$ is measurable with respect to $\mathcal{B}_{Q}$ and is integrable with respect to $\lambda_{Q}$.

2. (Nonnegativity)(ii) $0\le\Phi_{N}(y)$ for every $y\in\mathbb{R}^{n}$.

3. (Periodicity)(iii) $\Phi_{N}(y+k)=\Phi_{N}(y)$ for every $y\in\mathbb{R}^{n}$ and every $k\in\mathbb{Z}^{n}$.

4. (Mass)(iv) $\displaystyle\int_{\mathbb{T}^{n}}\Phi_{N}\,dx=1$, the integrand being $\Phi_{N}|_{Q}$.

5. (The far-field integral bound)(v) Let $\delta\in\mathbb{R}$ satisfy $0<\delta$ and $2\delta<1$, and let $K_{\delta}$ be the closed interval determined by $\delta$ and $1-\delta$. Let

$$E_{\delta}=\{\,y\in Q\ :\ y_{i}\in K_{\delta}\ \text{for some }i\in[n]\,\}.$$

Then $E_{\delta}\in\mathcal{B}_{Q}$ and, for every $N\in\mathbb{N}$, the map $\mathbf{1}_{E_{\delta}}\Phi_{N}|_{Q}$ is integrable with respect to $\lambda_{Q}$. Moreover there is a nonnegative real number $b_{\delta}$, depending on $\delta$ and $n$ but not on $N$, such that

$$\int_{\mathbb{T}^{n}}\mathbf{1}_{E_{\delta}}\Phi_{N}\,dx\le\frac{b_{\delta}}{N}\qquad\text{for every }N\in\mathbb{N},$$

the integrand being $\mathbf{1}_{E_{\delta}}\Phi_{N}|_{Q}$.