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 Mat. Sb., 2003, Volume 194, Number 11, Pages 17–64 (Mi msb779)

Multidimensional Tauberian theorems for Banach-space valued generalized functions

Yu. N. Drozhzhinov, B. I. Zavialov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Multidimensional Tauberian theorems for the standard averages of tempered Banach-space valued distributions are stated and proved. These results enable one to determine from the asymptotic behaviour of the averages the asymptotic behaviour of the generalized function itself. The role of the asymptotic scale in these results is performed by the class of regularly varying functions. Special attention is paid to averaging kernels such that several of their moments or linear combinations of moments vanish. Important in these results is the structure of the zero set of the Fourier transformations of the kernels in question.
The results so established are applied to the study of the asymptotic properties of solutions of the Cauchy problem for the heat equation in the class of tempered distributions, to the problem of the diffusion of a many-component gas, and to the problem of the absence of the phenomenon of compensation of singularities for holomorphic functions in tube domains over acute cones.

DOI: https://doi.org/10.4213/sm779

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English version:
Sbornik: Mathematics, 2003, 194:11, 1599–1646

Bibliographic databases:

UDC: 517.5
MSC: Primary 46F12, 40E05; Secondary 35K05, 32A40

Citation: Yu. N. Drozhzhinov, B. I. Zavialov, “Multidimensional Tauberian theorems for Banach-space valued generalized functions”, Mat. Sb., 194:11 (2003), 17–64; Sb. Math., 194:11 (2003), 1599–1646

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb779
• https://doi.org/10.4213/sm779
• http://mi.mathnet.ru/eng/msb/v194/i11/p17

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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. N. Drozhzhinov, B. I. Zavialov, “Quasi-asymptotics and spherical representations of generalized functions”, Doklady Mathematics, 70:2 (2004), 754–757
2. Yu. N. Drozhzhinov, B. I. Zavialov, “Asymptotically homogeneous generalized functions and boundary properties of functions holomorphic in tubular cones”, Izv. Math., 70:6 (2006), 1117–1164
3. Khrennikov A.Yu., Shelkovich V.M., “Distributional asymptotics and $p$-adic Tauberian and Shannon-Kotelnikov theorems”, Asymptot. Anal., 46:2 (2006), 163–187
4. Yu. N. Drozhzhinov, B. I. Zavialov, “Applications of Tauberian theorems in some problems in mathematical physics”, Theoret. and Math. Phys., 157:3 (2008), 1678–1693
5. Jasson Vindas, “Regularizations at the origin of distributions having prescribed asymptotic properties”, Integral Transforms & Special Functions, 22:4 (2011), 375
6. Pilipovic S. Vindas J., “Multidimensional Tauberian Theorems For Vector-Valued Distributions”, Publ. Inst. Math.-Beograd, 95:109 (2014), 1–28
7. A. L. Yakymiv, “A Tauberian theorem for multiple power series”, Sb. Math., 207:2 (2016), 286–313
8. Yu. N. Drozhzhinov, “Multidimensional Tauberian theorems for generalized functions”, Russian Math. Surveys, 71:6 (2016), 1081–1134
9. S. Pilipović, J. Vindas, “Tauberian class estimates for vector-valued distributions”, Sb. Math., 210:2 (2019), 272–296
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