Harmonic analysis associated with a discrete Laplacian

It is well known that the fundamental solution of

$${u_t}\left( {n,t} \right) = u\left( {n + 1,t} \right) - 2u\left( {n,t} \right) + u\left( {n - 1,t} \right),n \in \mathbb{Z},$$

with u(n, 0) = δ _nm for every fixed m ∈ Z is given by u(n, t) = e ^?2t I _n?m (2t), where I _k (t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series W _t f(n) = Σ _m∈Z e ^?2t I _n?m (2t)f(m). This formula allows us to analyze some operators associated with the discrete Laplacian using semigroup theory. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ? ^p (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. We also show that the Riesz transforms essentially coincide with the so-called discrete Hilbert transform defined by D. Hilbert at the beginning of the twentieth century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.