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 §pi, 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 §numbers. 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 §numbers, 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

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 §naturals. Then the following hold.

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

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

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

4. (Mass)¶ Let $N\in\mathbb{N}$. Then

5. (Far-field decay)¶ 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