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We characterize several classes of test functions, among them Björck's ultra-rapidly decaying test functions and the Gelfand-Shilov spaces of typeS, in terms of the decay of their short-time Fourier transform and in terms of their Gabor coefficients.

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The Schwartz space of rapidly decaying test functions is characterized by the decay of the short‐time Fourier transform or cross‐Wigner distribution. Then a version of Hardy's theorem is proved for the short‐time Fourier transform and for the Wigner distribution.

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