I try to understand a particular step on page two from the research paper "A multidimensional analog of a limit theorem of G. Szegö". https://iopscience.iop.org/article/10.1070/IM1975v009n06ABEH001523/meta
Problem Description:
Let $k\in \mathbb{Z}^m$ and $F(\lambda)\geq0$ $(\lambda \in [0,2\pi]^m)$ be a real function summable on $[0,2\pi]^m$ with Fourier coefficients $$ R(k) = \frac{1}{(2\pi)^m} \int_{[0,2\pi]^m} e^{-i\sum_{j=1}^m k_j\lambda_j} F(\lambda)d\lambda. $$
Let G be a bounded finitely connected domain in $\mathbb{R}^m$ with 2-smooth boundary $\partial G$ each connected component of which has positive Gaussian curvature. Let $T\in(0,\infty)$; let $G_T$ be the domain consisting of all the points Tx for x in G and $v_T$ the number of elements in $G'_T=G_T \cap\mathbb{Z}^m$.
Let $R_T$ be a convolution operator acting on the space $Y_T$ of all finite functions on $G'_T$, taking the function $\Psi(k)$, $k \in G'_T$, to the function $$ (R_T \Psi)(k)= \sum_{l\in G'_T}R(k-l)\Psi(l). $$
Problem: Computation of the trace of $R_T^n$.
Solution: Let $q=(k^1,...,k^n)$, $k^j \in \mathbb{Z}^m$. We define the function $$ H(q)=R(k^1-k^2)R(k^2-k^3)...R(k^n-k^1). $$ We introduce the subset $U_T=\{ G'_T \times (\mathbb{Z}^m)^{n-1}\} \setminus (G'_T)^n$ $\in (\mathbb{Z}^m)^n$ and the quantity $B_n(T)=\sum_{q\in U_T} H(q)$. Then $$ \text{trace} R_T^n= v_T S^{*n}(0)-B_n(T), $$ where * stands for the for operation of conjugation.
Question: My question is very simple how do I get to the solution.
I would be very pleased about any suggestions or tips for the solution.
