Frames of translates with prescribed fine structure in shift invariant spaces

TL;DR

This paper provides a criterion for constructing shift generated Bessel sequences with prescribed spectral and norm structures in finitely generated shift invariant spaces, including optimal sequences minimizing spectral spread.

Contribution

It introduces a simple criterion for the existence of shift generated Bessel sequences with prescribed fine structure and develops an eigensteps analogue for detailed sequence characterization.

Findings

Established a criterion for the existence of sequences with prescribed spectra and norms.

Developed an eigensteps-like framework for detailed sequence description.

Identified optimal sequences minimizing spectral spread using convex potentials.

Abstract

For a given finitely generated shift invariant (FSI) subspace $\cW\subset L^2(\R^k)$ we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences $E(\cF)$ induced by finite sequences of vectors $\cF\in \cW^n$ that have a prescribed fine structure i.e., such that the norms of the vectors in $\cF$ and the spectra of $S_{E(\cF)}$ is prescribed in each fiber of $\text{Spec}(\cW)\subset \T^k$. We complement this result by developing an analogue of the so-called sequences of eigensteps from finite frame theory in the context of SG Bessel sequences, that allows for a detailed description of all sequences with prescribed fine structure. Then, given $0<\alpha_1\leq \ldots\leq \alpha_n$ we characterize the finite sequences $\cF\in\cW^n$ such that $\|f_i\|^2=\alpha_i$, for $1\leq i\leq n$, and such that the fine spectral structure of the shift generated Bessel sequences $E(\cF)$ have minimal spread (i.e. we show the existence of optimal SG Bessel sequences with prescribed norms); in this context the spread of the spectra is measured in terms of the convex potential $P^\cW_\varphi$ induced by $\cW$ and an arbitrary convex function $\varphi:\R_+\rightarrow \R_+$.