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

holds for all . For a Gabor frame the

and its adjoint, the

are bounded linear operators. The

(We will often drop the subscript and denote the Gabor frame operator simply by ).

If constitutes a Gabor frame for , any function can be written as

where are the elements of the dual frame, given by . Equation (0.8) provides a constructive answer how to recover from its Gabor transform for given analysis window and how to compute the coefficients in the series expansion for given atom . The key is the corresponding dual frame .

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.