Rational discrete analytic functions on a rhombic lattice
Daniel Alpay, Zubayir Kazi, Mariana Tecalero, Dan Volok
TL;DR
This work develops a theory of rational discrete analytic (DA) functions on a rhombic lattice Λ by introducing a convolution product ⊙ and a DA polynomial basis built from shifted eigenfunctions. It defines forward and backward shift operators Z_+ and Z_- on the DA space, constructs the DA powers z^{(n)} via a Taylor-type expansion of eigenfunctions e_t, and establishes a complete realization-based framework for rational DA functions: f(z) = D + C(I − zA)^{−⊙} ⊙ (zB). The τ-map then identifies these ⊙-rational DA functions with classical rational matrix functions in t that have no poles in a lattice-dependent set P(Λ), enabling explicit products, inverses, and sums via realizations. The paper also provides forward-shift invariance criteria and includes an explicit example K_w(z) that yields a reproducing kernel, signaling potential Hilbert-space structures to be explored in future work. Overall, the results generalize discrete analytic rationality to rhombic lattices and integrate shift-operator methods with linear realization theory to obtain concrete algebraic operations on DA functions.
Abstract
A novel basis of discrete analytic polynomials on a rhombic lattice is introduced and the associated convolution product is studied. A class of discrete analytic functions that are rational with respect to this product is also described.