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Mat. Sb. (N.S.), 1986, Volume 130(172), Number 3(7), Pages 335–348 (Mi msb1879)  

This article is cited in 9 scientific papers (total in 9 papers)

Summability of the logarithm of an almost analytic function and a generalization of the Levinson–Cartwright theorem

A. L. Vol'berg, B. Jöricke


Abstract: This paper is devoted to a generalization of a classical inequality: let $f$ be bounded and analytic in the disk $D$; then $f\not\equiv0\Rightarrow\int_{\mathrm{Fr}\mathbf D}\log|f(e^{i\theta})| d\theta>-\infty$, in the case of nonanalytic functions $f$. More precisely, it is proved that if $f=f_1+f_2$, where $f_1$ is the boundary function of a function of bounded characteristic, and $f_2$ is a function in a quasianalytic class (defined by some condition of regularity of decrease of its Fourier coefficients), then $\int_{\mathrm{Fr}\mathbf D}\log|f(e^{i\theta})| d\theta>-\infty$. The proof of this result depends in an essential way on a theorem of Levinson and Cartwright. At the same time, the result strengthens the Levinson–Cartwright theorem.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 58:2, 337–349

Bibliographic databases:

UDC: 517.5
MSC: 30C80, 30D60, 30E25, 30D50
Received: 25.06.1985

Citation: A. L. Vol'berg, B. Jöricke, “Summability of the logarithm of an almost analytic function and a generalization of the Levinson–Cartwright theorem”, Mat. Sb. (N.S.), 130(172):3(7) (1986), 335–348; Math. USSR-Sb., 58:2 (1987), 337–349

Citation in format AMSBIB
\Bibitem{VolJor86}
\by A.~L.~Vol'berg, B.~J\"oricke
\paper Summability of the logarithm of an almost analytic function and a~generalization of the Levinson--Cartwright theorem
\jour Mat. Sb. (N.S.)
\yr 1986
\vol 130(172)
\issue 3(7)
\pages 335--348
\mathnet{http://mi.mathnet.ru/msb1879}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=865765}
\zmath{https://zbmath.org/?q=an:0644.30020}
\transl
\jour Math. USSR-Sb.
\yr 1987
\vol 58
\issue 2
\pages 337--349
\crossref{https://doi.org/10.1070/SM1987v058n02ABEH003107}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Volberg A., “A Criterion on a Subdomain of the Disk for its Harmonic Measure to Be Comparable with Lebesgue Measure”, Proc. Amer. Math. Soc., 112:1 (1991), 153–162  crossref  mathscinet  zmath  isi
    2. Dynkin E., “The Pseudoanalytic Extension”, J. Anal. Math., 60 (1993), 45–70  mathscinet  isi
    3. Jean Esterle, “Countable inductive limits of frechet algebras”, J Anal Math, 71:1 (1997), 195  crossref  mathscinet  zmath  isi
    4. J ESTERLE, A VOLBERG, “Sous-espaces invariants par translations bilatérales de certains espaces de Hilbert de suites quasi-analytiquement pondérées”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 326:3 (1998), 295  crossref
    5. Harlouchet I., “Closed Ideals of Certain Quasi-Analytic Beurling Algebras on the Unit Circle”, J. Math. Pures Appl., 79:9 (2000), 863–899  crossref  mathscinet  zmath  isi
    6. A. M. Gaisin, I. D. Latypov, “An estimate for the Dirichlet series with Fejér gaps on the real axis”, Siberian Math. J., 45:1 (2004), 53–68  mathnet  crossref  mathscinet  zmath  isi  elib
    7. A. M. Gaisin, “Dirichlet series with real coefficients that are unbounded on the positive half-axis”, Sb. Math., 198:6 (2007), 793–815  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. A. M. Gaisin, “Levinson's Condition in the Theory of Entire Functions: Equivalent Statements”, Math. Notes, 83:3 (2008), 317–326  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. St. Petersburg Math. J., 30:1 (2019), 1–13  mathnet  crossref  mathscinet  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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