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Multidimensional Tauberian theorems for vector-valued distributions

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Abstract
We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform M-phi(f)(x, y) = (f * phi(y))(x), (x, y) is an element of R-n x R+, with kernel phi(y) (t) = y(-n)phi(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on {x(0)} x R-m. In addition, we present a new proof of Littlewood's Tauberian theorem.
Keywords
FOURIER-SERIES, ORIGIN, CONVERGENCE, BEHAVIOR, TEMPERED DISTRIBUTIONS, GENERALIZED-FUNCTIONS, Laplace transform, wavelet transform, regularizing transforms, asymptotic behavior of generalized functions, PRIME NUMBER THEOREM, Abelian and Tauberian theorems, vector-valued distributions, quasiasymptotics, slowly varying functions

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Chicago
Pilipović, Stevan, and Jasson Vindas Diaz. 2014. “Multidimensional Tauberian Theorems for Vector-valued Distributions.” PUBLICATIONS DE L’INSTITUT MATHEMATIQUE-BEOGRAD 95 (109): 1–28.
APA
Pilipović, S., & Vindas Diaz, J. (2014). Multidimensional Tauberian theorems for vector-valued distributions. PUBLICATIONS DE L’INSTITUT MATHEMATIQUE-BEOGRAD, 95(109), 1–28.
Vancouver
1.
Pilipović S, Vindas Diaz J. Multidimensional Tauberian theorems for vector-valued distributions. PUBLICATIONS DE L’INSTITUT MATHEMATIQUE-BEOGRAD. 2014;95(109):1–28.
MLA
Pilipović, Stevan, and Jasson Vindas Diaz. “Multidimensional Tauberian Theorems for Vector-valued Distributions.” PUBLICATIONS DE L’INSTITUT MATHEMATIQUE-BEOGRAD 95.109 (2014): 1–28. Print.
@article{5662273,
  abstract     = {We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform M-phi(f)(x, y) = (f * phi(y))(x), (x, y) is an element of R-n x R+, with kernel phi(y) (t) = y(-n)phi(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on \{x(0)\} x R-m. In addition, we present a new proof of Littlewood's Tauberian theorem.},
  author       = {Pilipovi\'{c}, Stevan and Vindas Diaz, Jasson},
  issn         = {0350-1302},
  journal      = {PUBLICATIONS DE L'INSTITUT MATHEMATIQUE-BEOGRAD},
  keyword      = {FOURIER-SERIES,ORIGIN,CONVERGENCE,BEHAVIOR,TEMPERED DISTRIBUTIONS,GENERALIZED-FUNCTIONS,Laplace transform,wavelet transform,regularizing transforms,asymptotic behavior of generalized functions,PRIME NUMBER THEOREM,Abelian and Tauberian theorems,vector-valued distributions,quasiasymptotics,slowly varying functions},
  language     = {eng},
  number       = {109},
  pages        = {1--28},
  title        = {Multidimensional Tauberian theorems for vector-valued distributions},
  url          = {http://dx.doi.org/10.2298/PIM1409001P},
  volume       = {95},
  year         = {2014},
}

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