Wavelet expansions and asymptotic behavior of distributions

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.