Algebraic Structure and Orthogonality Properties of Rational Polynomial Phase Rings on Discrete Lattices
Abstract
We formalize the algebraic structure of rational polynomial phase sequences (discrete chirps) over the integer lattice ℤ. By constructing a convolution algebra generated by polynomial phases with coefficients in 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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BibTeX
@misc{smith2026algebraic,
title = {{Algebraic Structure and Orthogonality Properties of Rational Polynomial Phase Rings on Discrete Lattices}},
author = {Terence R. Smith},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.01650},
eprint = {2608.01650},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01650},
}
RIS
TY - UNPB
TI - Algebraic Structure and Orthogonality Properties of Rational Polynomial Phase Rings on Discrete Lattices
AU - Terence R. Smith
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.01650
UR - https://arxiv.org/abs/2608.01650
AB - We formalize the algebraic structure of rational polynomial phase sequences (discrete chirps) over the integer lattice ℤ. By constructing a convolution algebra generated by polynomial phases with coefficients in 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.
ER -