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Mat. Sb., 2006, Volume 197, Number 11, Pages 13–30 (Mi msb3788)  

This article is cited in 19 scientific papers (total in 19 papers)

Self-similar functions in $L_2[0,1]$ and the Sturm–Liouville problem with singular indefinite weight

A. A. Vladimirov, I. A. Sheipak

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The question of the asymptotic behaviour of the spectrum of the boundary value problem
\begin{equation*} -y"-\lambda\rho y=0, \qquad y(0)=y(1)=0, \end{equation*}
is considered, where $\rho$ is a function in $\mathring W_2^{-1}[0,1]$ with arithmetically self-similar primitive function. It is not assumed here that the weight $\rho$ has a constant sign. The theoretical results obtained are illustrated by the data of numerical calculations.
Bibliography: 10 titles.


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English version:
Sbornik: Mathematics, 2006, 197:11, 1569–1586

Bibliographic databases:

UDC: 517.984
MSC: Primary 34B25, 47B50; Secondary 47E05
Received: 16.06.2004 and 21.06.2006

Citation: A. A. Vladimirov, I. A. Sheipak, “Self-similar functions in $L_2[0,1]$ and the Sturm–Liouville problem with singular indefinite weight”, Mat. Sb., 197:11 (2006), 13–30; Sb. Math., 197:11 (2006), 1569–1586

Citation in format AMSBIB
\by A.~A.~Vladimirov, I.~A.~Sheipak
\paper Self-similar functions in $L_2[0,1]$ and the
Sturm--Liouville problem with singular indefinite weight
\jour Mat. Sb.
\yr 2006
\vol 197
\issue 11
\pages 13--30
\jour Sb. Math.
\yr 2006
\vol 197
\issue 11
\pages 1569--1586

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    This publication is cited in the following articles:
    1. A. A. Vladimirov, I. A. Sheipak, “Indefinite Sturm–Liouville Problem for Some Classes of Self-similar Singular Weights”, Proc. Steklov Inst. Math., 255 (2006), 82–91  mathnet  crossref  mathscinet  elib
    2. A. A. Vladimirov, “Calculating the eigenvalues of the Sturm–Liouville problem with a fractal indefinite weight”, Comput. Math. Math. Phys., 47:8 (2007), 1295–1300  mathnet  crossref  mathscinet  elib  elib
    3. A. A. Vladimirov, “On the oscillation theory of the Sturm–Liouville problem with singular coefficients”, Comput. Math. Math. Phys., 49:9 (2009), 1535–1546  mathnet  crossref  zmath  isi  elib  elib
    4. I. A. Sheipak, “Singular points of a self-similar function of spectral order zero: self-similar Stieltjes string”, Math. Notes, 88:2 (2010), 275–286  mathnet  crossref  crossref  mathscinet  isi
    5. A. A. Vladimirov, I. A. Sheipak, “Asymptotics of the Eigenvalues of the Sturm–Liouville Problem with Discrete Self-Similar Weight”, Math. Notes, 88:5 (2010), 637–646  mathnet  crossref  crossref  mathscinet  isi
    6. Sheipak I.A., “Spectrum of a Jacobi matrix with exponentially growing matrix elements”, Moscow Univ. Math. Bull., 66:6 (2011), 244–249  mathnet  crossref  mathscinet  zmath  elib  elib
    7. Nazarov A.I., Sheipak I.A., “Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes”, Bull London Math Soc, 44:1 (2012), 12–24  crossref  mathscinet  zmath  isi  elib
    8. N. V. Gaganov, I. A. Sheipak, “A boundedness criterion for the variations of self-similar functions”, Siberian Math. J., 53:1 (2012), 55–71  mathnet  crossref  mathscinet  isi
    9. A. A. Vladimirov, I. A. Shejpak, “Eigenvalue asymptotics of the problem of high odd order with dicrete self-similar weight”, St. Petersburg Math. J., 24:2 (2013), 263–273  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    10. Jonathan Eckhardt, Gerald Teschl, “Sturm-Liouville operators with measure-valued coefficients”, JAMA, 120:1 (2013), 151  crossref  mathscinet  zmath  isi
    11. A. A. Vladimirov, I. A. Sheipak, “On the Neumann Problem for the Sturm–Liouville Equation with Cantor-Type Self-Similar Weight”, Funct. Anal. Appl., 47:4 (2013), 261–270  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. J. V. Tikhanov, “On the Rate of Approximation of Singular Functions by Step Functions”, Math. Notes, 95:4 (2014), 530–543  mathnet  crossref  crossref  mathscinet  isi  elib
    13. N. V. Rastegaev, “On spectral asymptotics of the Neumann problem for the Sturm–Liouville equation with self-similar generalized Cantor type weight”, J. Math. Sci. (N. Y.), 210:6 (2015), 814–821  mathnet  crossref
    14. I. A. Sheipak, “Asymptotics of the Spectrum of a Differential Operator with the Weight Generated by the Minkowski Function”, Math. Notes, 97:2 (2015), 289–294  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. A. A. Vladimirov, “Nekotorye zamechaniya ob integralnykh kharakteristikakh vinerovskogo protsessa”, Dalnevost. matem. zhurn., 15:2 (2015), 156–165  mathnet  elib
    16. J. V. Tikhonov, I. A. Sheipak, “On the string equation with a singular weight belonging to the space of multipliers in Sobolev spaces with negative index of smoothness”, Izv. Math., 80:6 (2016), 1242–1256  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    17. A. S. Ivanov, A. M. Savchuk, “Trace of Order $(-1)$ for a String with Singular Weight”, Math. Notes, 102:2 (2017), 164–180  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    18. I. A. Sheipak, “O pokazatelyakh Geldera samopodobnykh funktsii”, Funkts. analiz i ego pril., 53:1 (2019), 67–78  mathnet  crossref  elib
    19. A. A. Vladimirov, “On a class of singular Sturm–Liouville problems”, Trans. Moscow Math. Soc., 80 (2019), 211–219  mathnet  crossref  elib
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