Wavelet expansions and asymptotic behavior of distributions |
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Authors: | Katerina Saneva |
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Institution: | a Faculty of Electrical Engineering and Information Technologies, University ‘Ss. Cyril and Methodius’, Karpos 2 bb, 1000 Skopje, Macedonia b Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281 Gebouw S22, B 9000 Gent, Belgium |
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Abstract: | We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line S0(R)⊂S(R) and its dual space , namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in . A characterization of boundedness and convergence in 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. |
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Keywords: | Orthogonal wavelets Wavelet coefficients Abelian theorems Tauberian theorems Distributions Quasiasymptotics Slowly varying functions Asymptotic behavior of generalized functions |
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