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 $1\le p,q\le \mathrm{\infty }$, a non-negative function $m(x,\omega )$ on ${\mathbb{ℝ}}^{2d}$ and a test function $g\in {𝒮}({\mathbb{ℝ}}^d)$, the modulation space $M_m^{p,q}({\mathbb{ℝ}}^d)$ is defined by

$M_m^{p,q}({\mathbb{ℝ}}^d)=\{f\in {{𝒮}}^{\prime }({\mathbb{ℝ}}^d):{\left(\underset{{\mathbb{ℝ}}^d}{\int }{(\underset{{\mathbb{ℝ}}^d}{\int }|V_{g}f(x,\omega ){|}^{p}m(x,\omega {)}^{p}dx)}^{q/p}d\omega \right)}^{1/q}<\mathrm{\infty }\}.$

In the above equation, $V_{g}f$ denotes the short-time Fourier transform of $f$ with respect to $g$ evaluated at $(x,\omega )$, namely

$V_{g}f(x,\omega )=\underset{{\mathbb{ℝ}}^d}{\int }f(t)\overline{g(t-x)}{e}^{-2\pi it\cdot \omega }dt={{ℱ}}_{\xi }^{-1}(\overline{\hat{g}(\xi )}\hat{f}(\xi +\omega ))(x).$

In other words, $f\in M_m^{p,q}({\mathbb{ℝ}}^d)$ is equivalent to $V_{g}f\in L_m^{p,q}({\mathbb{ℝ}}^{2d})$. The space $M_m^{p,q}({\mathbb{ℝ}}^d)$ is the same, independent of the test function $g\in {𝒮}({\mathbb{ℝ}}^d)$ chosen. The canonical choice is a Gaussian.

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

$M_{p,q}^s({\mathbb{ℝ}}^d)=\{f\in {{𝒮}}^{\prime }({\mathbb{ℝ}}^d):{(\sum _{k\in {\mathbb{\mathbb{Z}}}^d}⟨k{⟩}^{sq}\Vert {\psi }_k(D)f{\Vert }_p^q)}^{1/q}<\mathrm{\infty }\},⟨x⟩:=|x|+1$,

where $\{{\psi }_k\}$ is a suitable unity partition. If $m(x,\omega )=⟨\omega {⟩}^s$, then $M_{p,q}^s=M_m^{p,q}$.

Feichtinger's algebra

For $p=q=1$ and $m(x,\omega )=1$, the modulation space $M_m^{1,1}({\mathbb{ℝ}}^d)=M^1({\mathbb{ℝ}}^d)$ is known by the name Feichtinger's algebra and often denoted by $S_0$ for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators. $M^1({\mathbb{ℝ}}^d)$ is a Banach space embedded in $L^1({\mathbb{ℝ}}^d)\cap C_0({\mathbb{ℝ}}^d)$, and is invariant under the Fourier transform. 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. Fourier transform ${ℱ}$ is an automorphism on $M^{1,1}$.