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 Szeg\H{o} 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.