The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay

In the setting of The Real Numbers: Standing Notation and Background, let $\cos$ and $\sin$ be the cosine and sine functions from $\mathbb{R}$ to $\mathbb{R}$, let $\pi$ be the real number of The Number Pi (i), let $\mathbb{Z}$ be the set of integers, let $2=1+1$, and for a real number $t$ let $t^{2}=tt$ as in The Real Numbers: Standing Notation and Background (i). Natural numbers are read in $\mathbb{R}$ through the canonical map fixed there. Let $J=\{t\in\mathbb{R}:0\le t<1\}$ and let $(J,\mathcal{B}_{J},\lambda_{J})$ be the measure space introduced in The Integral over the Unit Cell of a Product of One-Variable Functions; for a map $w:\mathbb{R}\to\mathbb{R}$ we write $\int_{J}w\,d\lambda_{J}$ for the integral of the restriction $w|_{J}$, as in Cell Integrals of the Trigonometric Monomials, and $C_{m}$ for $m\in\mathbb{Z}$ denotes the map $t\mapsto\cos(2\pi mt)$ fixed there. Continuity means continuity relative to $\mathbb{R}$ formed with the metric $d_{\mathbb{R}}$ of The Real Numbers: Standing Notation and Background (i), and a closed interval is as defined there.

For $N\in\mathbb{N}$ let $F_{N}:\mathbb{R}\to\mathbb{R}$, the Fejer kernel of order $N$, be given by

$$F_{N}(t)=\frac{1}{N}\Bigl(N+2\sum_{m=1}^{N}(N-m)\cos(2\pi mt)\Bigr)\qquad(t\in\mathbb{R}),$$

the quotient by $N$, which is positive in $\mathbb{R}$ by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; finite sums are those of The Real Numbers: Standing Notation and Background (ii). Then the following hold.

1. (Regularity and periodicity)(i) Let $N\in\mathbb{N}$. Then $F_{N}$ is continuous on $\mathbb{R}$, its restriction to $J$ is $\lambda_{J}$-integrable, and

$$F_{N}(t+j)=F_{N}(t)\qquad\text{for every }t\in\mathbb{R}\text{ and every }j\in\mathbb{Z}.$$

2. (The kernel identity)(ii) Let $N\in\mathbb{N}$ and $t\in\mathbb{R}$. Then

$$N\,F_{N}(t)\bigl(\sin(\pi t)\bigr)^{2}=\bigl(\sin(N\pi t)\bigr)^{2}.$$

3. (Nonnegativity)(iii) Let $N\in\mathbb{N}$. Then $0\le F_{N}(t)$ for every $t\in\mathbb{R}$.

4. (Mass)(iv) Let $N\in\mathbb{N}$. Then

$$\int_{J}F_{N}\,d\lambda_{J}=1 .$$

5. (Far-field decay)(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 $1-\delta$. Then there is a positive real number $c_{\delta}$ such that

$$F_{N}(t)\le\frac{1}{N\,c_{\delta}}\qquad\text{for every }N\in\mathbb{N}\text{ and every }t\in K_{\delta}.$$