Structure of nonstationary Gabor frames and their dual systems

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. 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 necessary and sufficient condition for a pair of nonstationary Gabor frames to form dual frames, valid under mild restrictions. This condition is then applied in a simple setup, to prove the existence of dual pairs of nonstationary Gabor systems with coarser frequency sampling than allowed by previous results [3]. We also explore a connection to recent work of Christensen, Kim and Kim on Gabor frames with compactly supported window function.