Generalized Fourier series and shift-invariant subspaces

TL;DR

This paper explores the structure of principal shift-invariant subspaces of L^2(R) by relating them to weighted L^2 spaces on the torus, and investigates their generators using basis properties of trigonometric systems.

Contribution

It establishes a connection between shift-invariant subspaces and weighted norm spaces, providing new results on generators based on earlier basis property findings.

Findings

Characterization of shift-invariant subspaces as weighted L^2 spaces

Results on generators derived from basis properties of trigonometric systems

Extension of previous basis results to subspace generators

Abstract

A principal shift invariant subspace of $L^{2}(\IR)$ is isometric to a weighted norm space $L^{2}(\IT, w)$. Using results obtained earlier by the author on the basis properties of subsystems of the trigonometric system in the weighted norm spaces $L^{2}(\IT, w)$ we obtain corresponding results concerning generators of principal shift invariant subspaces.