Structure of nonstationary Gabor frames and their dual systems

TL;DR

This paper explores the structural properties of dual systems in nonstationary Gabor frames, revealing conditions under which inverse frame operators have Walnut-like representations and establishing criteria for dual frame pairs.

Contribution

It introduces new structural results for dual systems in nonstationary Gabor frames, including Walnut-like representations and duality conditions, especially with compactly supported windows.

Findings

Inverse nonstationary Gabor frame operators can have Walnut-like representations.

Canonical dual frames partially inherit the structure of original frames with specific differences.

Conditions for dual frame pairs are established and shown to be equivalent to duality under certain restrictions.

Abstract

We investigate the structural properties of dual systems for nonstationary Gabor frames. In particular, we prove that some inverse nonstationary Gabor frame operators admit a Walnut-like representation, i.e. the operator acting on a function can be described by weighted translates of that function, even when the original frame operator is not diagonal. In this case, which only occurs when compactly supported window functions are used, the canonical dual frame partially inherits the structure of the original frame, with differences that we describe in detail. Moreover, we determine a sufficient condition for a pair of nonstationary Gabor frames to form dual frames. The equivalence of this condition to the duality of the involved systems is shown under some weak restrictions. It is then applied in a simple setup, to prove the existence of dual pairs of nonstationary Gabor systems with non-diagonal frame operator. A discussion of the results, restricted to the classical case of regular Gabor systems, precedes the statement of the general results. Here, we also explore a connection to recent work of Christensen, Kim and Kim on Gabor frames with compactly supported window function.