Modulation space

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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

In the above equation, denotes the short-time Fourier transform of with respect to evaluated at , namely

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 .

Feichtinger's algebra[edit]

For and , the modulation space is known by the name Feichtinger's algebra and often denoted by for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators. is a Banach space embedded in , and is invariant under the Fourier transform. It is for these and more properties that is a natural choice of test function space for time-frequency analysis. Fourier transform is an automorphism on .

References[edit]

  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.