Modulation space explained

Modulation spaces^[1] are a family of Banach spaces defined by the behavior of the short-time Fourier transform withrespect 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 inf ty , a non-negative function m(x,\omega) on R ^{2d} and a test function g \in l{S}( R ^{d)} , the modulation space M ^{d)} _{m( R} is defined byM ^{d)} _{m( R} = \left\{ f\in l{S}'( R ^{d) : \left(\int} _{R ^{d}} \left(\int _{R ^{d}} |V ^{p} _{gf(x,\omega)|} m(x,\omega) ^{p} dx\right) ^{q/p} d\omega\right) ^{1/q} < inf ty\right\}.

In the above equation,

V _{gf} denotes the short-time Fourier transform of f with respect to g evaluated at (x,\omega) , namelyV _{gf(x,\omega)=\int} ^{R ^{d}} f(t)\overline{g(t - x)}e ^{-2\pi} dt = l{F} ^{-1} _{\xi} (\overline{\hat{g}(\xi)}\hat{f}(\xi + \omega))(x).

In other words,

f\in M ^{d)} _{m( R} is equivalent to V _{gf\in} L ^{2 d} _{m( R} ) . The space M ^{d)} _{m( R} is the same, independent of the test function g \in l{S}( R ^{d)} chosen. The canonical choice is a Gaussian.

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

M ^{s} _{p,q} ( R ^{d)} = \left\{ f\in l{S}'( R ^{d) : \left(\sum} _{k\in Z ^{d}} \langle k \rangle ^{sq} \|\psi _{k(D)f\|} ^{q\right)} ^{p} ^{1/q} < inf ty\right\}, \langle x\rangle: = |x| +1 ,where \{\psi _{k\}} is a suitable unity partition. If m(x,\omega) = \langle \omega\rangle ^{s} , then M ^{s} _{p,q} = M ^{p,q} _{m} .

Feichtinger's algebra

For

p = q =1 and m(x,\omega) = 1 , the modulation space M ^{d)} _{m( R} = M ^{1(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(R} ^{d)} is a Banach space embedded in L ^{1(R} ^{d)} \cap C ^{d)} _{0 ( R} , and is invariant under the Fourier transform. It is for these and more properties that M ^{1(R} ^{d)} is a natural choice of test function space for time-frequency analysis. Fourier transform l{F} is an automorphism on M ^{1,1} .

Notes and References

1. https://books.google.com/books?id=yTnTBwAAQBAJ Foundations of Time-Frequency Analysis
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.