The Reproducing Identity for the Fejer Kernels of 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 $\mathbb{R}^{n}$, whose points are read as maps on $[n]$ with real values and whose difference $x-y$ is formed coordinatewise, and the integer lattice $\mathbb{Z}^{n}$ are the ones fixed there. Let $\mathbb{Z}$ be the set of integers; natural numbers are read in $\mathbb{R}$ through the canonical map fixed in The Real Numbers: Standing Notation and Background (i), and are then integers by claim 1 of Arithmetic, Order, Discreteness and Intervals of the Integers. Let $|t|$ denote the absolute value of a real number $t$. Let $\phi_{m}$ for $m\in\mathbb{Z}$ be the one-dimensional trigonometric maps and $e_{k}$ for $k\in\mathbb{Z}^{n}$ the trigonometric system on $\mathbb{T}^{n}$; for $N\in\mathbb{N}$ let $F_{N}$ be the Fejer kernel of order $N$ and $\Phi_{N}$ the Fejer kernel of the torus of order $N$, so that $\Phi_{N}(y)=\prod_{i=1}^{n}F_{N}(y_{i})$ for $y\in\mathbb{R}^{n}$. Finite sums and finite products are those of The Real Numbers: Standing Notation and Background (ii) and Finite Product Notation in a Field, and a sum indexed by a nonempty finite set is that of Sum over a Finite Index Set.

Let $N\in\mathbb{N}$, and let $2N+1$ denote the natural number $N+(N+1)$. For $a\in[2N+1]$ the real number $a-N-1$ is an integer, by claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers, and we put

$$w_{N,a}=1-\frac{|a-N-1|}{N},$$

the quotient being by $N$, which is positive in $\mathbb{R}$, hence invertible, by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Let $[2N+1]^{n}$ be the set of $n$-tuples in $[2N+1]$, that is, of maps $a:[n]\to[2N+1]$, with components written $a_{i}$; the initial segment $[2N+1]$ is nonempty and finite by claim 1 of Properties of a Sum over a Finite Index Set, so $[2N+1]^{n}$ is nonempty and finite by claim 3 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets. For $a\in[2N+1]^{n}$ let $k(a)$ be the point of $\mathbb{R}^{n}$ with coordinates $k(a)_{i}=a_{i}-N-1$ for $i\in[n]$, which exists by claim 2 of Euclidean Points as Tuples of Real Numbers and lies in $\mathbb{Z}^{n}$ by Lattice-Periodic Functions and the Periodic Function Classes (i), and put

$$W_{N,a}=\prod_{i=1}^{n}w_{N,a_{i}} .$$

Then the following hold.

1. (One variable)(i) For all $t,s\in\mathbb{R}$,

$$F_{N}(t-s)=\sum_{a=1}^{2N+1}w_{N,a}\,\phi_{a-N-1}(t)\,\phi_{a-N-1}(s).$$

2. (The product kernel)(ii) For all $x,y\in\mathbb{R}^{n}$,

$$\Phi_{N}(x-y)=\sum_{a\in[2N+1]^{n}}W_{N,a}\,e_{k(a)}(x)\,e_{k(a)}(y).$$

3. (Symmetry)(iii) $\Phi_{N}(x-y)=\Phi_{N}(y-x)$ for all $x,y\in\mathbb{R}^{n}$.