Modulation space

Modulation spaces 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, is identical to a certain modulation space and has become a widely used space of test functions for time-frequency analysis. Modulation spaces 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, 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 ≤ ∞ {displaystyle 1leq p,qleq infty } , a non-negative function m ( x , ω ) {displaystyle m(x,omega )} on R 2 d {displaystyle mathbb {R} ^{2d}} and a test function g ∈ S ( R d ) {displaystyle gin {mathcal {S}}(mathbb {R} ^{d})} , the modulation space M m p , q ( R d ) {displaystyle M_{m}^{p,q}(mathbb {R} ^{d})} is defined by In the above equation, V g f {displaystyle V_{g}f} denotes the short-time Fourier transform of f {displaystyle f} with respect to g {displaystyle g} evaluated at ( x , ω ) {displaystyle (x,omega )} , namely In other words, f ∈ M m p , q ( R d ) {displaystyle fin M_{m}^{p,q}(mathbb {R} ^{d})} is equivalent to V g f ∈ L m p , q ( R 2 d ) {displaystyle V_{g}fin L_{m}^{p,q}(mathbb {R} ^{2d})} . The space M m p , q ( R d ) {displaystyle M_{m}^{p,q}(mathbb {R} ^{d})} is the same, independent of the test function g ∈ S ( R d ) {displaystyle gin {mathcal {S}}(mathbb {R} ^{d})} chosen. The canonical choice is a Gaussian. We also have a Besov-type definition of modulation spaces as follows. where { ψ k } {displaystyle {psi _{k}}} is a suitable unity partition. If m ( x , ω ) = ⟨ ω ⟩ s {displaystyle m(x,omega )=langle omega angle ^{s}} , then M p , q s = M m p , q {displaystyle M_{p,q}^{s}=M_{m}^{p,q}} . For p = q = 1 {displaystyle p=q=1} and m ( x , ω ) = 1 {displaystyle m(x,omega )=1} , the modulation space M m 1 , 1 ( R d ) = M 1 ( R d ) {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 S 0 {displaystyle S_{0}} for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators. M 1 ( R d ) {displaystyle M^{1}(mathbb {R} ^{d})} is a Banach space embedded in L 1 ( R d ) ∩ C 0 ( R d ) {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 M 1 ( R d ) {displaystyle M^{1}(mathbb {R} ^{d})} is a natural choice of test function space for time-frequency analysis. Fourier transform F {displaystyle {mathcal {F}}} is an automorphism on M 1 , 1 {displaystyle M^{1,1}} .