The theory of frames is due to Duffin and Schaeffer [DS52], and was introduced in the context of nonharmonic Fourier series only six years after Gabor published his paper. Despite this fact one has to say that frame analysis has become popular much later in sampling theory, time-frequency analysis and wavelet theory, ignited by the papers [DGM86] and [Dau90].
A system is a Gabor frame or Weyl-Heisenberg frame for , if there exist two constants such that
If constitutes a Gabor frame for , any function can be written as
A detailed analysis of Gabor frames brings forward some features that are basic for a further understanding of Gabor analysis. Most of these features are not shared by other frames such as wavelet frames.