Modulation space

Modulation spaces^[1] 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^[2], 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 $1\leq p,q \leq \infty$, a non-negative function m(x,ω) on $\mathbb{R}^{2d}$ and a test function $g \in \mathcal{S}(\mathbb{R}^d)$, the modulation space $M^{p,q}_m(\mathbb{R}^d)$ is defined by

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

In the above equation, V_gf denotes the short-time Fourier transform of f with respect to g evaluated at (x,ω). In other words, $f\in M^{p,q}_m(\mathbb{R}^d)$ is equivalent to $V_gf\in L^{p,q}_m(\mathbb{R}^{2d})$. It should be noted, that the space $M^{p,q}_m(\mathbb{R}^d)$ is the same, independent of the test function $g \in \mathcal{S}(\mathbb{R}^d)$ chosen. The canonical choice is a Gaussian.

Feichtinger's algebra

For p = q = 1 and m(x,ω) = 1, the modulation space $M^{1,1}_m(\mathbb{R}^d) = M^1(\mathbb{R}^d)$ is known by the name Feichtinger's algebra and often denoted by S₀ for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators. $M^1(\mathbb{R}^d)$ is invariant under Fourier transform and a Banach space embedded in $L^1(\mathbb{R}^d) \cap C_0(\mathbb{R}^d)$. It is for these and more properties that $M^1(\mathbb{R}^d)$ is a natural choice of test function space for time-frequency analysis.

References

1. ^ Foundations of Time-Frequency Analysis by Karlheinz Gröchenig
2. ^ H. Feichtinger. "On a new Segal algebra" Monatsh. Math. 92:269–289, 1981.