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arXiv · 2608.01650

Algebraic Structure and Orthogonality Properties of Rational Polynomial Phase Rings on Discrete Lattices

Terence R. Smith
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Abstract

We formalize the algebraic structure of rational polynomial phase sequences (discrete chirps) over the integer lattice $\mathbb{Z}$. By constructing a convolution algebra generated by polynomial phases with coefficients in $\mathbb{Q}$, we establish a rigorous operational calculus for discrete difference equations. We prove an exact orthogonality relation for these sequences over finite periodic windows, yielding a generalized discrete Fourier isometry. Furthermore, we demonstrate that the continuum limit of this discrete framework can be rigorously defined via the inductive limit of finite-dimensional $C^*$-algebras (an AF-algebra). This framework provides a rigorous algebraic substrate for finite-mode lattice approximations in quantum mechanics and the construction of mutually unbiased bases in finite-dimensional quantum information theory.

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Terence R. Smith. 2026-08-03. Algebraic Structure and Orthogonality Properties of Rational Polynomial Phase Rings on Discrete Lattices. https://arxiv.org/abs/2608.01650

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