Fundamental properties of Gabor frames

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 $ \{g_{m,n}\} = \{T_{na} M_{mb} g\}$ is a Gabor frame or Weyl-Heisenberg frame for $ \Ltsp(\R)$, if there exist two constants $ A,B >0$ such that

$\displaystyle A \Vert f\Vert^2 \le \sum_{m,n \in \Z} \vert\langle f, g_{m,n}\rangle \vert^2 \le B \Vert f\Vert^2$ (5)

holds for all $ f \in \Ltsp(\R)$. For a Gabor frame $ \{g_{m,n}\}$ the analysis mapping (also called Gabor transform) $ T_g$, given by

$\displaystyle T_g: f \rightarrow \{\langle f , g_{m,n}\rangle\}_{m,n}$ (6)

and its adjoint, the synthesis mapping (also called Gabor expansion) $ T^*_g$, given by

$\displaystyle \{c_{m,n}\} \rightarrow \sum_{m,n \in \Z} c_{m,n} \, g_{m,n} \qquad \{c_{m,n}\} \in \ltZ$ (7)

are bounded linear operators. The Gabor frame operator $ S_g$ is defined by $ S_g = T_g^* T_g$. Explicitly,

$\displaystyle S_g f = \sum_{m,n \in \Z} \langle f,g_{m,n}\rangle \, g_{m,n}\,.$ (8)

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

If $ \{g_{m,n}\}$ constitutes a Gabor frame for $ \Ltsp(\R)$, any function $ f \in \Ltsp(\R)$ can be written as

$\displaystyle f = \sum_{m,n} \langle f, g_{m,n} \rangle \,\gamma_{m,n} = \sum_{m,n} \langle f, \gamma_{m,n} \rangle \,g_{m,n}$ (9)

where $ \gamma_{m,n}$ are the elements of the dual frame, given by $ \gamma_{m,n} = S^{-1} g_{m,n}$. Equation (0.8) provides a constructive answer how to recover $ f$ from its Gabor transform $ \{\langle f, g_{m,n}\rangle\}_{m,n \in \Z}$ for given analysis window $ g$ and how to compute the coefficients in the series expansion $ f = \sum c_{m,n} g_{m,n}$ for given atom $ g$. The key is the corresponding dual frame $ \{S^{-1} g_{m,n}\}_{m,n \in \Z}$.

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.