Discrete harmonic extensions and a discrete Hilbert transform

The classical Hilbert transform of a function on the real line arises as the boundary value of the function’s conjugate-harmonic extension. In this article, harmonic extensions to the upper half-integer lattice Z × N Z × N double struck upper Z times double struck upper N are constructed for a given boundary sequence on the integers; these serve as discrete analogs of the Poisson and conjugate-Poisson integrals. The construction is characterized by: (i) discrete harmonicity with respect to a two-dimensional Laplacian, (ii) a Cauchy–Riemann system, and (iii) boundary values involving a discrete Hilbert transform. This new discrete Hilbert transform is compared to the Riesz–Titchmarsh transform, and Lebesgue bounds are proved. The construction is also extended to the higher dimensional case. These results provide a discrete harmonic-analytic model analogous to the classical theory.