Continuously Frame-Convertible Sequences

TL;DR

This paper introduces a relaxed class of sequences in Hilbert spaces that can be continuously mapped to Parseval frames, enabling similar reconstruction formulas with broader applicability.

Contribution

It characterizes these sequences via analysis operators, provides examples including non-complete sequences, and classifies measures related to exponential systems.

Findings

Sequences can be continuously mapped to Parseval frames even if not complete.

Norm-based criteria determine when Schauder sequences have this property.

Classification of measures on the torus for exponential systems with this property.

Abstract

Frame theory provides a robust method for recovering vectors in a Hilbert space from inner product data, though the associated decomposition formula can be computationally demanding. We relax the frame condition by studying sequences that can be continuously mapped to Parseval frames, yielding a similar reconstruction formula. We characterize such sequences in terms of their analysis operators, without reference to any continuous mapping. We present examples, including sequences that are not complete and those containing no frame sequence. We also give norm-based criteria for when unconditional Schauder sequences and finite unions of bounded unconditional Schauder sequences admit this property. Finally, we classify finite Borel measures on the torus for which the standard exponential system has this property and forms a Riesz Fischer sequence.