On R-duals of type III in Hilbert spaces

TL;DR

This paper characterizes R-duals of type III in Hilbert spaces, extending existing constructions using antiunitary involutions and providing spectral measure representations for certain classes.

Contribution

It introduces new characterizations and a systematic construction method for R-duals of type III, expanding the theoretical framework in Hilbert space frame theory.

Findings

New characterizations of R-duals of type III

Extended construction method using antiunitary involutions

Spectral measure representation for specific classes

Abstract

Following work by Casazza, Kutyniok and Lammers, and extensions by Stoeva and Christensen, we provide some novel characterizations of R-dual sequences of type III in Hilbert spaces. We systematically extend the construction procedure by basing it on a choice of an antiunitary involution. For certain classes of R-duals of type III, we derive a representation of the associated frame operator in terms of spectral measures.