We define and study the Ridgelet transform of (Lizorkin) distributions. We establish connections with the Radon and wavelet transforms.

We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line S 0 (R) ⊂ S(R) and its dual space S 0 (R), namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in S 0 (R). A characterization of boundedness and convergence in S 0 (R) is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients.

We study the short-time Fourier transform on the space K-1'(R-n) of distributions of exponential type. We give characterizations of K-1'(R-n) and some of its subspaces in terms of modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform

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