On discrete holomorphic Paley–Wiener spaces and sampling on the square lattice

We consider a reproducing kernel Hilbert space of discrete entire functions on the square lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb Z^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> inspired by the classical Paley–Wiener space of entire functions of exponential growth in the complex plane. For such space we provide a Paley–Wiener type characterization and a sampling result.