Gabor frame

One may be interested in Gabor frames and its related theory if one looks further into the frame framework. First, denote a lattice by $\mathrm{\Lambda }=A{\mathbb{Z}}^{2d}$, where $A$ is an invertible matrix, and let $\pi (\xi ,\varphi )f=e^{2\pi i\xi x}f(x-\varphi )$

Definition.

Let $g\mathrm{\in }{L}^{\mathrm{2}}\mathit{}\mathrm{(}{\mathrm{R}}^d\mathrm{)}$ be a nonzero window, and let $\lambda \mathrm{\in }\mathrm{\Lambda }$, then

$$G(g,\lambda )=\{\pi (\lambda )g:\lambda \in \mathrm{\Lambda }\}$$

is a Gabor system. If $G\mathit{}\mathrm{(}g\mathrm{,}\lambda \mathrm{)}$ is a frame, it’s called a Gabor frame for $L^{\mathrm{2}}\mathit{}\mathrm{(}{\mathrm{R}}^d\mathrm{)}$

Supose now that one wants to look at a more general framework, and work with functions in $L^2({ℝ}^d,{ℂ}^n)$. Then the definition above generalises to

Definition.

Let $𝐠\mathrm{\in }{L}^{\mathrm{2}}\mathit{}\mathrm{(}{\mathrm{R}}^d\mathrm{,}{\mathrm{C}}^n\mathrm{)}$ be a nonzero window and let $\lambda \mathrm{\in }\mathrm{\Lambda }$, then

$$𝑮(𝒈,\lambda )=\{\pi (\lambda )𝒈:\lambda \in \mathrm{\Lambda }\}$$

is a Gabor super-frame if the frame inequalities hold, where

$$\pi (\xi ,\varphi )𝒈=e^{2\pi ıx\cdot \xi }(g_1(x-\varphi ),g_2(x-\varphi ),\mathrm{\dots },g_n(x-\varphi ))$$

and for $𝐟\mathrm{,}𝐡\mathrm{\in }{L}^{\mathrm{2}}\mathit{}\mathrm{(}{\mathrm{R}}^d\mathrm{,}{\mathrm{C}}^n\mathrm{)}$

$${⟨𝒇,𝒉⟩}_{L^2({ℝ}^d,{ℂ}^n)}=\sum _{i=1}^n{⟨f_i,h_i⟩}_{L^2({ℝ}^d)}$$