Frame-related Sequences in Chains and Scales of Hilbert Spaces

TL;DR

This paper investigates how frame-related properties of sequences in Hilbert space scales propagate across different spaces, revealing that certain properties are preserved while others cannot, especially in non-trivial scales.

Contribution

It provides a detailed analysis of the propagation of frame properties in Hilbert space chains and establishes limitations on dual-frame existence across different scale spaces.

Findings

Upper semi-frame property persists in larger spaces.

Lower semi-frame property persists in smaller spaces.

A sequence cannot be a frame for two different spaces in a non-trivial scale.

Abstract

Frames for Hilbert spaces are interesting for mathematicians but also important for applications e.g. in signal analysis and in physics. Both in mathematics and physics it is natural to consider a full scale of spaces, and not only a single one. In this paper, we study how certain frame-related properties, as completeness or the property of being a (semi-)frame, of a certain sequence in one of the spaces propagate to other spaces in a scale. We link that to the properties of the respective frame-related operators, like analysis or synthesis. We start with a detailed survey of the theory of Hilbert chains. Using a canonical isomorphism the properties of frame sequences are naturally preserved between different spaces. We also show that some results can be transferred if the original sequence is considered, in particular that the upper semi-frame property is kept in larger spaces, while the lower one to smaller ones. This leads to a negative result: a sequence can never be a frame for two Hilbert spaces of the scale if the scale is non-trivial, i.e. spaces are not equal.