Abstract. It is well-known that the fundamental solution of ut(n, t) = u(n + 1, t) − 2u(n, t) + u(n − 1, t), n ∈ Z, with u(n, 0) = δnm for every fixed m ∈ Z, is given by u(n, t) = e −2t In−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 seriesBy using semigroup theory, this formula allows us to analyze some operators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ℓ p (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essentially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX 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.

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size h > 0for u, f : Z h → R, 0 < s < 1, is performed. The pointwise nonlocal formula for (−∆ h ) s u and the nonlocal discrete mean value property for discrete s-harmonic functions are obtained. We observe that a characterization of (−∆ h ) s as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Hölder spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincaré inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution)which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy-Littlewood-Sobolev inequality for (−∆ h ) −s .As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as h → 0 in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of h under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian (−∆) s U = F, in R, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of h.2010 Mathematics Subject Classification. Primary: 35R11, 49M25. Secondary: 35K05, 65N15. Key words and phrases. Nonlocal discrete diffusion equations, fractional discrete Laplacian, regularity and extension problem, Sobolev and Poincaré inequalities, error of approximation, semidiscrete heat equation.

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A transplantation theorem for Jacobi series proved by Muckenhoupt is reinvestigated by means of a suitable variant of Calderón-Zygmund operator theory. An essential novelty of our paper is weak type (1,1) estimate for the Jacobi transplantation operator, located in a fairly general weighted setting. Moreover, L p estimates are proved for a class of weights that contains the class admitted in Muckenhoupt's theorem.

Let J denote the Bessel function of order . The functions y␣ r2y␤ r2y1r2 Ž 1r 2 .x J x , n s 0, 1, 2, . . . , form an orthogonal system in ␣q␤q2 nq1. In this paper we analyze the range of p, ␣ , and ␤ for which the Fourier series with respect to this system converges in the p ŽŽ . ␣ . L 0, ϱ , x dx -norm. Also, we describe the space in which the span of the system is dense and we show some of its properties. Finally, we study the almost everywhere convergence of the Fourier series for functions in such spaces. ᮊ Academic Press