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Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 4, Pages 47–118 (Mi izv395)  

This article is cited in 14 scientific papers (total in 14 papers)

Tauberian theorems for generalized functions with values in Banach spaces

Yu. N. Drozhzhinov, B. I. Zavialov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We state and prove Tauberian theorems of a new type. In these theorems we give sufficient conditions under which the values of a generalized function (distribution) that are assumed to lie in a locally convex topological space actually belong to some narrower (Banach) space. These conditions are stated in terms of “general class estimates” for the standard average of this generalized function with a fixed kernel belonging to a space of test functions.
The applications of these theorems are based, in particular, on the fact that asymptotical (and some other) properties of the generalized functions under investigation can be described in terms of membership of certain Banach spaces. We apply these theorems to the study of asymptotic properties of solutions of the Cauchy problem for the heat equation in the class of generalized functions of small growth (tempered distributions), and to the study of Banach spaces of Besov–Nikol'skii type.

DOI: https://doi.org/10.4213/im395

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English version:
Izvestiya: Mathematics, 2002, 66:4, 701–769

Bibliographic databases:

UDC: 517.5
MSC: 46F12, 40E05, 44A15
Received: 30.08.2001

Citation: Yu. N. Drozhzhinov, B. I. Zavialov, “Tauberian theorems for generalized functions with values in Banach spaces”, Izv. RAN. Ser. Mat., 66:4 (2002), 47–118; Izv. Math., 66:4 (2002), 701–769

Citation in format AMSBIB
\by Yu.~N.~Drozhzhinov, B.~I.~Zavialov
\paper Tauberian theorems for generalized functions with values in Banach spaces
\jour Izv. RAN. Ser. Mat.
\yr 2002
\vol 66
\issue 4
\pages 47--118
\jour Izv. Math.
\yr 2002
\vol 66
\issue 4
\pages 701--769

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    This publication is cited in the following articles:
    1. Drozhzhinov Yu. N., Zav'yalov B.I., “On one multidimensional Tauberian theorem for generalized functions taking values in Banach spaces”, Dokl. Math., 68:1 (2003), 30–33  mathnet  mathscinet  zmath  isi  elib
    2. Yu. N. Drozhzhinov, B. I. Zavialov, “Multidimensional Tauberian theorems for Banach-space valued generalized functions”, Sb. Math., 194:11 (2003), 1599–1646  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Vindas J., Pilipović S., “Structural theorems for quasiasymptotics of distributions at the origin”, Math. Nachr., 282:11 (2009), 1584–1599  crossref  mathscinet  zmath  isi  elib  scopus
    5. Vindas J., Estrada R., “Measures and the distributional $\phi$-transform”, Integral Transforms Spec. Funct., 20:3-4 (2009), 325–332  crossref  mathscinet  zmath  isi  elib  scopus
    6. Vindas J., Estrada R., “On the Support of Tempered Distributions”, Proceedings of the Edinburgh Mathematical Society, 53:Part 1 (2010), 255–270  crossref  mathscinet  zmath  isi  scopus
    7. Jasson Vindas, Stevan Pilipović, Dušan Rakić, “Tauberian Theorems for the Wavelet Transform”, J Fourier Anal Appl, 2010  crossref  mathscinet  isi  scopus
    8. Ricardo Estrada, “The set of singularities of regulated functions in several variables”, Collect. Math, 63:3 (2011), 351  crossref  mathscinet  isi  scopus
    9. Estrada R., Vindas J., “A General Integral”, Diss. Math., 2012, no. 483, 5+  mathscinet  isi
    10. Pavel Dimovski, Stevan Pilipović, Jasson Vindas, “New distribution spaces associated to translation-invariant Banach spaces”, Monatsh Math, 2014  crossref  mathscinet  scopus
    11. Pilipovic S., Vindas J., “Multidimensional Tauberian Theorems For Vector-Valued Distributions”, Publ. Inst. Math.-Beograd, 95:109 (2014), 1–28  crossref  mathscinet  zmath  isi  scopus
    12. A. L. Yakymiv, “A Tauberian theorem for multiple power series”, Sb. Math., 207:2 (2016), 286–313  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. Yu. N. Drozhzhinov, “Multidimensional Tauberian theorems for generalized functions”, Russian Math. Surveys, 71:6 (2016), 1081–1134  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. S. Pilipović, J. Vindas, “Tauberian class estimates for vector-valued distributions”, Sb. Math., 210:2 (2019), 272–296  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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