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

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

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

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

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 $g\in {\mathcal {S}}(\mathbb {R} ^{d})$ chosen. The canonical choice is a Gaussian.

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

$M_{p,q}^{s}(\mathbb {R} ^{d})=\left\{f\in {\mathcal {S}}'(\mathbb {R} ^{d})\ :\ \left(\sum _{k\in \mathbb {Z} ^{d}}\langle k\rangle ^{sq}\|\psi _{k}(D)f\|_{p}^{q}\right)^{1/q}<\infty \right\},\langle x\rangle :=|x|+1$,

where $\{\psi _{k}\}$ is a suitable unity partition. If $m(x,\omega )=\langle \omega \rangle ^{s}$, 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 ${\mathcal {F}}$ is an automorphism on $M^{1,1}$.