Spaces of test functions via the STFT
The aim of the present paper is to investigate the time-frequency behavior of functions in Björk's spaces of ultra-rapidly decreasing test functions and in the Gelfand-Shilov spaces. The short-time Fourier transform (STFT) of a function or distribution \(f\) on \(\mathbb{R}^d\) with respect to the non-zero window \(g\) is defined as \[ ({\mathcal V}_gf)(x,\xi) = \int_{\mathbb{R}^d}f(t) \overline{g(t-x)}e^{-2\pi i \xi t} \, dt = \langle f, M_{\xi}T_xg\rangle \] where \(T_xg(t) = g(t-x)\) is the translation operator and \(M_{\xi}f(t) = e^{2\pi i \xi t}f(t)\) is the modulation operator. Let \(\sigma\) be an increasing, continuous and concave function on \([0,\infty)\) with \(\sigma (0) = 0\), \(\int_0^{\infty}\frac{\sigma(r)}{1+r^2}\,dr < \infty\) and \(\sigma(r) \geq c + d\log(1+r)\) for some \(c\in \mathbb{R}\) and \(d>0.\) For \(\omega (t):=\sigma(| t| )\) (\(t\in \mathbb{R}^d\)), the Björk space \(S_{\omega}(\mathbb{R}^d)\) of ultra-rapidly decreasing test functions is the Fréchet space of all infinitely differentiable functions \(f\) with the property that \(\|e^{\lambda \omega}D^p f\|_{L^{\infty}} < \infty\) and \(\|e^{\lambda \omega}D^p \widehat{f}\|_{L^{\infty}} < \infty\) for every \(\lambda >0\) and \(p\in \mathbb{N}_0.\) Theorem. Let \(g\in S_{\omega}(\mathbb{R}^d)\setminus \{0\}\) be fixed. Then for \(f\in S'_{\omega}(\mathbb{R}^d)\), the following are equivalent: (1) \(f\in S_{\omega}(\mathbb{R}^d),\) (2) for all \(\lambda >0\) there is \(C_{\lambda}>0\) such that \(| ({\mathcal V}_gf)(x,\xi)| \leq C_{\lambda}e^{-\lambda(\omega(x) + \omega(\xi))}\), (3) \({\mathcal V}_gf \in S_{\omega}(\mathbb{R}^{2d}).\) A similar result for the Schwartz space \(S(\mathbb{R}^d)\) was obtained by the same authors in [J. London Math. Soc. II. Ser. 63, No. 1, 205--214 (2001; Zbl 1106.46021)]. The previous theorem is reformulated using modulation spaces and, as a consequence, the authors characterize the functions in \(S_{\omega}(\mathbb{R}^d)\) by means of their coefficients with respect to a Gabor frame in \(L^2(\mathbb{R}^d).\) Moreover, a description of the Gelfand-Shilov spaces of type \({\mathcal S}\) in terms of the decay of the STFT is obtained.
scientific article; zbMATH DE number 2111347