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

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

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

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

We also have a Besov-type definition of modulation spaces as follows.[3]

${\displaystyle 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 ${\displaystyle \{\psi _{k}\}}$ is a suitable unity partition. If ${\displaystyle m(x,\omega )=\langle \omega \rangle ^{s}}$, then ${\displaystyle M_{p,q}^{s}=M_{m}^{p,q}}$.

Feichtinger's algebra

For ${\displaystyle p=q=1}$ and ${\displaystyle m(x,\omega )=1}$, the modulation space ${\displaystyle M_{m}^{1,1}(\mathbb {R} ^{d})=M^{1}(\mathbb {R} ^{d})}$ is known by the name Feichtinger's algebra and often denoted by ${\displaystyle S_{0}}$ for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators. ${\displaystyle M^{1}(\mathbb {R} ^{d})}$ is a Banach space embedded in ${\displaystyle L^{1}(\mathbb {R} ^{d})\cap C_{0}(\mathbb {R} ^{d})}$, and is invariant under the Fourier transform. It is for these and more properties that ${\displaystyle M^{1}(\mathbb {R} ^{d})}$ is a natural choice of test function space for time-frequency analysis. Fourier transform ${\displaystyle {\mathcal {F}}}$ is an automorphism on ${\displaystyle M^{1,1}}$.

References

1. Foundations of Time-Frequency Analysis by Karlheinz Gröchenig
2. H. Feichtinger. "On a new Segal algebra" Monatsh. Math. 92:269–289, 1981.
3. B.X. Wang, Z.H. Huo, C.C. Hao, and Z.H. Guo. Harmonic Analysis Method for Nonlinear Evolution Equations. World Scientific, 2011.