On the structure of finitely generated shift-invariant subspaces

TL;DR

This paper characterizes finitely generated shift-invariant subspaces with g-minimal generators, provides an algorithm for coefficient determination in Fourier representations, and offers norm estimates using an orthogonalization approach.

Contribution

It introduces a new characterization, an explicit algorithm, and norm estimates for generators of finitely generated shift-invariant subspaces.

Findings

Characterization of shift-invariant subspaces with g-minimal generators

Algorithm for Fourier coefficient determination

Norm estimates for coefficients

Abstract

A characterization of finitely generated shift-invariant subspaces is given when generators are g-minimal. An algorithm is given for the determination of the coefficients in the well known representation of the Fourier transform of an element of the finitely generated shift-invariant subspace as a linear combination of Fourier transformations of generators. An estimate for the norms of those coefficients is derived. For the proof a sort of orthogonalization procedure for generators is used which reminds the well known Gram-Schmidt orthogonalization process.