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
, a non-negative function
on
and a test function
, the modulation space
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\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b22fee105dcef3d7402bdb56e0b38968259e3e0)
In the above equation,
denotes the short-time Fourier transform of
with respect to
evaluated at
, 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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d440c1aba10dc01bd2a4ab9ba5f5f260e0677e1a)
In other words,
is equivalent to
. The space
is the same, independent of the test function
chosen. The canonical choice is a Gaussian.
We also have a Besov-type definition of modulation spaces as follows.[3]
,
where
is a suitable unity partition. If
, then
.
- ^ Foundations of Time-Frequency Analysis by Karlheinz Gröchenig
- ^ H. Feichtinger. "On a new Segal algebra" Monatsh. Math. 92:269–289, 1981.
- ^ B.X. Wang, Z.H. Huo, C.C. Hao, and Z.H. Guo. Harmonic Analysis Method for Nonlinear Evolution Equations. World Scientific, 2011.