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 ≤ p , q ≤ ∞ , a non-negative function m ( x , ω ) on R 2 d and a test function g ∈ S ( R d ) , the modulation space M m p , q ( R d ) is defined by
M m p , q ( R d ) = { f ∈ S ′ ( R d ) : ( ∫ R d ( ∫ R d | V g f ( x , ω ) | p m ( x , ω ) p d x ) q / p d ω ) 1 / q < ∞ } . In the above equation, V g f denotes the short-time Fourier transform of f with respect to g evaluated at ( x , ω ) , namely
V g f ( x , ω ) = ∫ R d f ( t ) g ( t − x ) ¯ e − 2 π i t ⋅ ω d t = F ξ − 1 ( g ^ ( ξ ) ¯ f ^ ( ξ + ω ) ) ( x ) . In other words, f ∈ M m p , q ( R d ) is equivalent to V g f ∈ L m p , q ( R 2 d ) . The space M m p , q ( R d ) is the same, independent of the test function g ∈ S ( 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 ( R d ) = { f ∈ S ′ ( R d ) : ( ∑ k ∈ Z d ⟨ k ⟩ s q ∥ ψ k ( D ) f ∥ p q ) 1 / q < ∞ } , ⟨ x ⟩ := | x | + 1 ,
where { ψ k } is a suitable unity partition. If m ( x , ω ) = ⟨ ω ⟩ s , then M p , q s = M m p , q .
For p = q = 1 and m ( x , ω ) = 1 , the modulation space M m 1 , 1 ( R d ) = 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 ) ∩ C 0 ( R d ) , 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 F is an automorphism on M 1 , 1 .
