Modulation space

Modulation spaces are a family of Banach spaces defined by the behavior of the short-time Fourier transform with respect to a test function from the Schwartz space. They were originally proposed by Hans Georg Feichtinger and are recognized to be the right kind of function spaces for time-frequency analysis. Feichtinger's algebra, while originally introduced as a new Segal algebra, is identical to a certain modulation space and has become a widely used space of test functions for time-frequency analysis.

Modulation spaces are defined as follows. For [math]\displaystyle{ 1\leq p,q \leq \infty }[/math], a non-negative function [math]\displaystyle{ m(x,\omega) }[/math] on [math]\displaystyle{ \mathbb{R}^{2d} }[/math] and a test function [math]\displaystyle{ g \in \mathcal{S}(\mathbb{R}^d) }[/math], the modulation space [math]\displaystyle{ M^{p,q}_m(\mathbb{R}^d) }[/math] is defined by

[math]\displaystyle{ M^{p,q}_m(\mathbb{R}^d) = \left\{ f\in \mathcal{S}'(\mathbb{R}^d)\ :\ \left(\int_{\mathbb{R}^d}\left(\int_{\mathbb{R}^d} |V_gf(x,\omega)|^p m(x,\omega)^p dx\right)^{q/p} d\omega\right)^{1/q} \lt \infty\right\}. }[/math]

In the above equation, [math]\displaystyle{ V_gf }[/math] denotes the short-time Fourier transform of [math]\displaystyle{ f }[/math] with respect to [math]\displaystyle{ g }[/math] evaluated at [math]\displaystyle{ (x,\omega) }[/math], namely

[math]\displaystyle{ V_gf(x,\omega)=\int_{\mathbb{R}^d}f(t)\overline{g(t-x)}e^{-2\pi it\cdot \omega}dt=\mathcal{F}^{-1}_{\xi}(\overline{\hat{g}(\xi)}\hat{f}(\xi+\omega))(x). }[/math]

In other words, [math]\displaystyle{ f\in M^{p,q}_m(\mathbb{R}^d) }[/math] is equivalent to [math]\displaystyle{ V_gf\in L^{p,q}_m(\mathbb{R}^{2d}) }[/math]. The space [math]\displaystyle{ M^{p,q}_m(\mathbb{R}^d) }[/math] is the same, independent of the test function [math]\displaystyle{ g \in \mathcal{S}(\mathbb{R}^d) }[/math] chosen. The canonical choice is a Gaussian.

We also have a Besov-type definition of modulation spaces as follows.

[math]\displaystyle{ M^s_{p,q}(\mathbb{R}^d) = \left\{ f\in \mathcal{S}'(\mathbb{R}^d)\ :\ \left(\sum_{k\in\mathbb{Z}^d} \langle k \rangle^{sq} \|\psi_k(D)f\|_p^q\right)^{1/q} \lt \infty\right\}, \langle x\rangle:=|x|+1 }[/math],

where [math]\displaystyle{ \{\psi_k\} }[/math] is a suitable unity partition. If [math]\displaystyle{ m(x,\omega)=\langle \omega\rangle^s }[/math], then [math]\displaystyle{ M^s_{p,q}=M^{p,q}_m }[/math].

Feichtinger's algebra

For [math]\displaystyle{ p=q=1 }[/math] and [math]\displaystyle{ m(x,\omega) = 1 }[/math], the modulation space [math]\displaystyle{ M^{1,1}_m(\mathbb{R}^d) = M^1(\mathbb{R}^d) }[/math] is known by the name Feichtinger's algebra and often denoted by [math]\displaystyle{ S_0 }[/math] for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators. [math]\displaystyle{ M^1(\mathbb{R}^d) }[/math] is a Banach space embedded in [math]\displaystyle{ L^1(\mathbb{R}^d) \cap C_0(\mathbb{R}^d) }[/math], and is invariant under the Fourier transform. It is for these and more properties that [math]\displaystyle{ M^1(\mathbb{R}^d) }[/math] is a natural choice of test function space for time-frequency analysis. Fourier transform [math]\displaystyle{ \mathcal{F} }[/math] is an automorphism on [math]\displaystyle{ M^{1,1} }[/math].