Orthonormal dilations of non-tight frames

TL;DR

This paper develops dilation theorems for non-tight frames with structured generation, extending prior results for Parseval frames and wavelets to broader classes of frames and identifying conditions for orthonormal dilations.

Contribution

It generalizes existing dilation theorems to non-tight frames generated by unitary groups and projective representations, and characterizes when frame wavelets can be dilated to orthonormal wavelets.

Findings

Extended dilation theorems to non-tight frames with additional structure.

Identified the optimal class of frame wavelets for orthonormal dilation.

Generalized previous results for Parseval frames and wavelets.

Abstract

We establish dilation theorems for non-tight frames with additional structure, i.e., frames generated by unitary groups of operators and projective unitary representations. This generalizes previous dilation results for Parseval frames due to Han and Larson and Gabardo and Han. We also extend the dilation theorem for Parseval wavelets, due to Dutkay, Han, Picioroaga, and Sun , by identifying the optimal class of frame wavelets for which dilation into an orthonormal wavelet is possible.