Hermite function description of Feichtinger's space \(S_{0}\) (Q2503131)

One says that \(f\in L^2(\mathbb{R})\) belongs to the space \(S_0\) provided its continuous Gabor transform is integrable, that is, \[ \| f\| _{S_0} =\int_{-\infty}^\infty \int_{-\infty}^\infty | \langle f, g_{x,y}\rangle | \, dx\, dy<\infty \] where \(g_{x,y}(t)= 2^{1/4} \exp (-\pi(t-x)^2+2\pi i y t)\). For \(\gamma\in \mathbb{R}\), let \(C(\gamma)\) denote those tempered distributions such that \(\sum_{n=1}^\infty | \langle f,\, h_n\rangle| (1+n)^\gamma <\infty\) where \(h_n\) denotes the \(n\)-th Hermite function which is, up to a normalizing constant, \(e^{\pi x^2} d^n (e^{-2\pi x^2})/dx^n\). It is shown that \(C(1/4)\subset S_0 \subset C(-1/4)\), while \(C(\gamma) \nsubseteq S_0\) if \(\gamma<1/4\) and \(S_0\nsubseteq C(\gamma)\) if \(\gamma>-1/4\). That \(C(1/4)\subset S_0\) is a consequence of Stirling's formula, while \(S_0\subset C(-1/4)\) follows from applying the inversion formula for the continuous Gabor transform. That \(S_0\nsubseteq C(\gamma)\) for \(\gamma>-1/4\) is proved as a consequence of Mehler's formula, while \(C(\gamma)\nsubseteq S_0\) for \(\gamma<1/4\) is proved from the fact that the Hermite functions \(h_n\) are concentrated on concentric annuli, with little overlap for largely different values of \(n\).