Analysis Seminar: Harmonic extensions on ℤ × ℕ and a discrete Hilbert transform

Abstract: 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 talk, harmonic extensions to the upper half-integer lattice ℤ × 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. We also extend the constructions to harmonic extensions on higher-dimensional lattices. These results provide a discrete harmonic-analytic model analogous to the classical theory.

Host: Alan Chang