Spectral Factorization and Lattice Geometry

TL;DR

This paper establishes conditions under which certain trigonometric polynomials can be approximated by squared magnitudes of polynomials with frequencies constrained by lattice projections, using advanced spectral and Diophantine techniques.

Contribution

It generalizes the Fejer-Riesz spectral factorization to higher dimensions and applies lattice geometry and Diophantine approximation to spectral factorization problems.

Findings

Derived conditions for spectral approximation using lattice projections.

Extended spectral factorization techniques to two dimensions.

Connected lattice geometry with spectral factorization through Diophantine approximation.

Abstract

We obtain conditions for a trigonometric polynomial t of one variable to equal or be approximated by |p|^2 where p has frequencies in a Bohr set of integers obtained by projecting lattice points in the open planar region bounded by the lines y = alpha*x +- beta where |beta| leq 1/4 and alpha is either rational or irrational with Liouville-Roth constant larger than 2. We derive and use a generalization of the Fejer-Riesz spectral factorization lemma in one dimension, an approximate spectral factorization in two dimensions, the modular group action on the integer lattice, and Diophantine approximation.