Discrete Tori and Trigonometric Sums

We prove a discrete analogue of the Poisson summation formula.


Introduction
The well know Poisson summation formula says that, for any positive real t , It can be proved by using the heat kernel p S t (x, y) on the unit circle S as follows. For the trace of the heat operator acting in L 2 (S), there are two expressions as follows: where λ j is the sequence of all the eigenvalues of the Laplace operator = d 2 dx 2 on S counted with multiplicity, and Comparing (1.2) and (1.3), using that that the sequence λ j consists of the numbers k 2 , k ∈ Z, and that is the Gauss-Weierstrass function, one obtains (1.1) (see, for example, [5,Exercise 10.18]).
Similar ideas have been widely used in the literature for obtaining various trace formulas and estimates of eigenvalues of Riemannian manifolds, for example, in [1,3,4], etc. In the framework of graphs we mention [2] where the above idea was applied to the heat kernels p T t (x, y) on discrete tori T in Z n and, hence, a certain analogue of the Poisson summation formula was obtained.
In this paper we also work with discrete tori but use a discrete time heat kernel q s (x, y), s ∈ Z + , instead of the one with a continuous time t ∈ R + . In fact, q s (x, y) is the transition density of a simple random walk on the graph in question. As a result, we obtain explicit formulas for some trigonometric sums that seems to be new.
Our results are stated in Theorem 3.2 and Corollary 3.4. To illustrate them, let us present them for 2-dimensional discrete tori. For any 2 × 2 integer matrix M and for any non-negative integer s, set The lattice 7Z 2 (double lines), the lattice AZ 2 (single lines) and the torus T A (shaded) Consider a 2 × 2 integer matrix A with m := det A>1. Then the lattice AZ 2 contains mZ 2 so that the quotient is well defined (in the sense of groups) and can be regarded as a discrete torus. Note that T A contains m vertices.

Example 1.2 Consider the matrix
The structure of this paper is as follows. In Sect. 2 we have collected all necessary information about the Markov operators on weighted graph and their heat kernels, including the heat kernels on Cartesian products and quotients of graphs. These facts are rather elementary but they are hardly available in the literature in this concise form. Section 3 contains the main results mentioned above, their proofs, and examples.

Weighted Graphs
We briefly outline some fact from [6] about heat kernels on weighted graphs. Let be a locally finite graph where we denote by also the set of vertices of this graph.
We write x ∼ y if the vertices x, y of are connected by an edge in . Let μ xy be a symmetric non-negative function on pairs x y of vertices such that μ xy > 0 ⇔ x ∼ y. and assume in what follows that μ (x) > 0 for all x ∈ (that is, each vertex has at least 1 edge).
Consider a Markov operator P = P acting on functions f : → R as follows: It is clear that P f ≥ 0 if f ≥ 0 and P1 = 1. It follows that P acts in any space l r ( , μ) with r ∈ [1, ∞] and satisfies the norm-bound P ≤ 1. Besides, P is self-adjoint in l 2 ( , μ).
The weight μ xy is called simple if μ xy = 1 for all x ∼ y. In this case, μ (x) = deg (x) and For any s ∈ Z + the power P s is well defined, and the sequence {P s } s≥0 is a reversible Markov chain on . It is easy to see that The function q s (x, y) is called the discrete time heat kernel or the transition density of the Markov chain {P s } .

Product of Regular Graphs
A graph is called d-regular if any vertex has exactly d neighbors, that is, Let j n j=1 be a finite sequence of graphs. Consider their Cartesian product The vertices of are n-tuples x j ∼ y j and x k = y k for all k = j.
The edges x ∼ y in are defined by the following rule: Let us endow all the graphs j and with a simple weight. We have then for the Markov operator P on Let us consider the Markov operator P j on j as acting also on functions f (x) on along the component x j , so that It follows that Since all the operators P j commute on , we obtain that, for any s ∈ Z + , it follows that

Quotient of Graphs
Let ( , μ) be a weighted graph with μ (x) > 0 so that the Markov operator P is well defined. Let G be a group of weighted graph automorphisms of , that is, Then the vertex weight μ (x) is also G-invariant. It follows that the operator P commutes with G, that is, Consequently, also q s (x, y) commutes with G, that is, Consider the quotient Q = /G that consists of the equivalence classes [x] of vertices x ∈ under the equivalent relation x ≡ y mod G ⇔ x = gy for some g ∈ G.
The quotient Q has a natural weight: However, the weight μ Q may be not simple because the G-orbit of y may have more than 1 vertex adjacent to x.
Observe that always Any G-periodic function f on can be regarded as a function on Q by Clearly, P f is also G-periodic. Let us verify that In what follows we simplify notation by writing x instead of [x] when this does not cause confusion.

The Heat Kernel on Z n
It is known that the transition density q Z s (x, y) of a simple random walk on Z is given by Setting where the summation indices s 1 , ..., s n satisfy in addition s i ≥ k i and s i ≡ k i mod 2.

Heat Kernels on Discrete Tori
Let us fix some integer valued n × n matrix M with m := det M > 1.
We regard MZ n as an additive group that acts on Z n by shifts. Consider a discrete torus that is a finite graph with m vertices. Let μ be the weight on T that comes from the simple weight of Z n by (2.4). By (2.5), we have μ (x) = 2n for any x ∈ T .

Eigenfunctions on Discrete Tori
The following function is an eigenfunction of P Z n for any w ∈ R n : Indeed, we have where {e k } is a canonical basis in R n . Hence, we obtain Note that the functions f w and f w are equal if and only of w = w mod Z n so that we can assume that w ∈ R n /Z n . Consider a lattice W := M * −1 Z n /Z n and a torus (2.10).

Moreover, the family { f w } w∈W forms an orthogonal basis in l 2 (T , μ) .
Proof To prove the first claim, it suffices to verify that Let the columns of M be u 1 , ..., u n . Then for y = e k we obtain My = u k so that w, u k = z k for some z k ∈ Z. The matrix of this linear system is M * , whence which finishes the proof of the first claim.
The fact that f w is an eigenfunction of P T follows from (2.6) and the fact that f w is an eigenfunction of P Z n as was verified above.
Let us verify that the family { f w } w∈W is orthogonal For all w = w , we have where w = w − w . Since w is non-zero as an element of the torus W , the eigenfunction f w is orthogonal to the eigenfunction f 0 = 1 because 0 is known to be a simple eigenvalue of P T . Hence, f w ⊥ f w as claimed.
Since the family { f w } w∈W is linearly independent and the number of elements in this family is equal to det M * = m, it follows that this family forms an orthogonal basis in l 2 (T , μ).

Main Result
As above, let us fix an integer valued n × n matrix M with m := det M > 1.
For any non-negative integer s, set Now we can state and prove our main result. Proof Since α w with w ∈ W are the eigenvalues of P T , we obtain using (2.11) (3.6) Substituting the value of α w from (3.1), we obtain (3.5).
It is convenient to rewrite (3.3) in the form where, for any v ∈ Z n and s ∈ Z + , Observe that the numbers C s (v) do not depend on M. By (3.7), the number C s (M) is determined by the vertices v of the lattice MZ n lying in the l 1 -ball in Z n of radius s (see Fig. 3). It is clear from (3.8) that if |v| ≡ s mod 2 then C s (v) = 0.
Consequently, the summation in (3.7) can be restricted to those v with |v| = s mod 2.
In the case n = 2 Theorem 3.2 can be reformulated as follows. By In this case, also the converse is true.
Hence, (3.11) follows from (3.5). Funding Open Access funding enabled and organized by Projekt DEAL.
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