Time-frequency transforms with Poisson kernel modulation

In this paper, a nonlinear modulation [Formula: see text] and a frequency-varying dilation [Formula: see text] both with Poisson kernel are introduced. Two classes of time-frequency atoms [Formula: see text] are designed from a basic atom [Formula: see text] in the Schwartz class [Formula: see text] acted upon by three operators: translation, nonlinear modulation and dilation. Two time-frequency transformations [Formula: see text] are constructed based on the above designed time-frequency atoms, where [Formula: see text] maps [Formula: see text] into [Formula: see text] with Lebesgue measure while [Formula: see text] maps [Formula: see text] into [Formula: see text] with Haar measure. The corresponding inversion formulae are established and the reproducing kernel Hilbert space property of the images of [Formula: see text] is proved. This strategy offers a unified understanding of dilation frequency and Fourier (modulation) frequency.