Hardy spaces of discrete holomorphic functions on the upper half-lattice

We develop a theory of Hardy spaces $H^p$ of discrete holomorphic functions on the upper half-lattice, within the classical framework of discrete holomorphicity on the square lattice.

We prove Cauchy and Poisson reproducing formulas, establish a boundary norm identity, and obtain Paley--Wiener type characterizations for these spaces.

In the Hilbert space case, we describe the associated reproducing kernel and Szegő projection, and we compare the discrete theory with the classical Hardy space on the upper half-plane through a family of discrete holomorphic approximants of classical $H^2$-functions.

We also prove duality results for $H^p$, $1<p<\infty$, establish uniqueness and sampling results on horizontal lines, and introduce Bergman-type spaces, comparing two natural weighted scales.