Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 

 

Abel and Tauberian theorems for integrals


Authors: A. F. Grishin and I. V. Poedintseva
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 3.
Journal: St. Petersburg Math. J. 26 (2015), 357-409
MSC (2010): Primary 40E05; Secondary 30D20
DOI: https://doi.org/10.1090/S1061-0022-2015-01343-5
Published electronically: March 20, 2015
MathSciNet review: 3289177
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A new method is suggested for obtaining Abel and Tauberian Theorems for integrals of the form $ \int _0^\infty K\big (\frac {t}{r}\big )\,d\mu (t)$. It is based on properties of limit sets for measures. Accordingly, a version of Azarin's cluster set theory for Radon measures on the half-line $ (0,\infty )$ is created. Theorems of new sort are proved, in which the asymptotic behavior of the above integrals is described in terms of cluster sets for $ \mu $. With the use of these results and a stronger version (also proved in the paper) of Karleman's well-known analytic continuation lemma, the second Tauberian theorem by Wiener is refined considerably.


References [Enhancements On Off] (What's this?)

  • 1. B. Ya. Levin, Distribution of zeros of entire functions, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian). MR 0087740
  • 2. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871
  • 3. H. S. A. Potter, The mean values of certain Dirichlet series, II, Proc. London Math. Soc. (2) 47 (1940), 1–19. MR 0005141, https://doi.org/10.1112/plms/s2-47.1.1
  • 4. V. S. Azarin, Asymptotic behavior of subharmonic functions of finite order, Mat. Sb. (N.S.) 108(150) (1979), no. 2, 147–167, 303 (Russian). MR 525835
  • 5. David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
  • 6. G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
  • 7. Jacob Korevaar, Tauberian theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 329, Springer-Verlag, Berlin, 2004. A century of developments. MR 2073637
  • 8. G. Valiron, Sur les fonctions entières d’ordre nul et d’ordre fini et en particulier les fonctions à correspondance régulière, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3) 5 (1913), 117–257 (French). MR 1508338
  • 9. A. F. Grishin and T. I. Malyutina, On the proximate order, Complex Analysis and Mathematical Physics, Krasnoyarsk, 1998, pp. 10-24. (Russian)
  • 10. N. Bourbaki, Éléments de mathématique. Fasc. XIII. Livre VI: Intégration. Chapitres 1, 2, 3 et 4: Inégalités de convexité, Espaces de Riesz, Mesures sur les espaces localement compacts, Prolongement d’une mesure, Espaces 𝐿^{𝑝}, Deuxième édition revue et augmentée. Actualités Scientifiques et Industrielles, No. 1175, Hermann, Paris, 1965 (French). MR 0219684
  • 11. Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • 12. N. S. Landkof, \cyr Osnovy sovremennoĭ teorii potentsiala, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0214795
  • 13. V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR 0121520
  • 14. Vladimir Azarin, Growth theory of subharmonic functions, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. MR 2463743
  • 15. V. S. Azarin and V. B. Giner, The structure of cluster sets of entire and subharmonic functions, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 38 (1982), 3–12, 126 (Russian). MR 686069
  • 16. A. F. Grishin and T. I. Malyutina, Density functions, Mat. Fiz. Anal. Geom. 7 (2000), no. 4, 387–414 (Russian, with English, Russian and Ukrainian summaries). MR 1811086
  • 17. A. Chouigui, Subharmonic function with extremely nonregular growth, Visnik Kharkiv. Univ. Ser. Mat. Prikl. Mat. Mech. 56 (2006), 80-85. (Russian)
  • 18. A. F. Grishin and T. I. Malyutina, New formulas for indicators of subharmonic functions, Mat. Fiz. Anal. Geom. 12 (2005), no. 1, 25–72 (Russian, with English, Russian and Ukrainian summaries). MR 2135424
  • 19. A. F. Grishin, The simplest Tauberian theorem, Mat. Zametki 74 (2003), no. 2, 221–229 (Russian, with Russian summary); English transl., Math. Notes 74 (2003), no. 1-2, 212–219. MR 2023764, https://doi.org/10.1023/A:1025052123976
  • 20. Norbert Wiener, The Fourier integral and certain of its applications, dover Publications, Inc., New York, 1959. MR 0100201
  • 21. M. A. Evgrafov, \cyr Analiticheskie funktsii, Second revised and enlarged edition, Izdat. “Nauka”, Moscow, 1968 (Russian). MR 0239053
  • 22. T. Carleman, L’Intégrale de Fourier et Questions que s’y Rattachent, Publications Scientifiques de l’Institut Mittag-Leffler, vol. 1, Uppsala, 1944 (French). MR 0014165
  • 23. V. P. Gurariĭ, Group methods of commutative harmonic analysis, Current problems in mathematics. Fundamental directions, Vol. 25, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 4–303 (Russian). MR 982753
  • 24. Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Bd. 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
  • 25. B. I. Korenbljum, A generalization of Wiener’s Tauberian theorem and harmonic analysis of rapidly increasing functions, Trudy Moskov. Mat. Obšč. 7 (1958), 121–148 (Russian). MR 0101455

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 40E05, 30D20

Retrieve articles in all journals with MSC (2010): 40E05, 30D20


Additional Information

A. F. Grishin
Affiliation: Department of Mathematics and Mechanics, V. N. Kazarin Kharkov National University, pl. Svobody 4, Kharkov 61022, Ukraine
Email: grishin@univer.kharkov.ua

I. V. Poedintseva
Affiliation: Department of Mathematics and Mechanics, V. N. Kazarin Kharkov National University, pl. Svobody 4, Kharkov 61022, Ukraine
Email: Irina.V.Poedintseva@univer.kharkov.ua

DOI: https://doi.org/10.1090/S1061-0022-2015-01343-5
Keywords: Valiron's proximate order, Radon measure, Azarin's cluster set for a measure, Azarin's regular measure, Wiener Tauberian theorem
Received by editor(s): September 5, 2013
Published electronically: March 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society