Closure of dilates of shift-invariant subspaces

TL;DR

This paper investigates the properties of shift-invariant subspaces under dilation, establishing equivalences between completeness and spectral conditions, and extends results to wavelet Fourier transforms.

Contribution

It introduces new spectral conditions characterizing the closure of dilates of shift-invariant subspaces and extends these results to $A$-reducing spaces and wavelet Fourier transforms.

Findings

Completeness is equivalent to spectral conditions at the origin.

The origin is a point of $A^*$-approximate continuity of the spectral function.

The Fourier transform of semiorthogonal tight frame wavelets is also $A^*$-approximately continuous.

Abstract

Let $V$ be any shift-invariant subspace of square summable functions. We prove that if for some $A$ expansive dilation $V$ is $A$-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of $V$, among them the origin is a point of $A^*$-approximate continuity of the spectral function if we assume this value to be one. We present our results also in the more general setting of $A$-reducing spaces. We also prove that the origin is a point of $A^*$-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero.