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Wavelet expansions and asymptotic behavior of distributions. (English) Zbl 1196.42031

Author’s abstract: We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line \(\mathcal S_0(\mathbb R)\subset \mathcal S (\mathbb R)\) and its dual space \(\mathcal S'_0(\mathbb R)\), namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in \(\mathcal S'_0(\mathbb R)\). A characterization of boundedness and convergence in \(\mathcal S'_0(\mathbb R)\) 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.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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