Convex potentials and optimal shift generated oblique duals in shift invariant spaces

TL;DR

This paper extends convex potentials to shift-invariant spaces, characterizing tight frames as minimizers and analyzing the spectral structure of optimal oblique duals with norm constraints.

Contribution

It introduces a new framework for convex potentials in shift-invariant spaces and characterizes optimal oblique duals through spectral and geometric analysis.

Findings

Convex potentials detect tight frames as minimizers.

Spectral analysis of shift generated oblique duals.

Optimality of water-filling construction in this context.

Abstract

We introduce an extension of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in $L^2(\R^k)$. We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.