Differential Equations

, Volume 55, Issue 1, pp 8–23 | Cite as

Convergence of the Spectral Expansion in the Eigenfunctions of a Fourth-Order Differential Operator

  • V. M. KurbanovEmail author
  • Kh. R. Godzhaeva
Ordinary Differential Equations


We study the convergence of spectral expansions of functions of the class W p 1 (G), p ≥ 1, G = (0, 1), in the eigenfunctions of an ordinary differential operator of even order with integrable coefficients. Sufficient conditions for absolute and uniform convergence are obtained and the rate of uniform convergence of these expansions on the interval ̅G is found.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Naimark, M.A., Lineinye differentsial’nye operatory (Linear Differential Operators), Moscow: Nauka, 1969.Google Scholar
  2. 2.
    Lazetic, N.L., On uniform convergence of spectral expansions and their derivatives arising by self-adjoint extensions of a one-dimensional Schrödinger operator, Publ. Inst. Math. Nouv. Ser., 2001, vol. 69 (83), pp. 59–77.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Lazetic, N.L., On uniform convergence on closed intervals of spectral expansions and their derivatives for functions from W1 p, Mat. Vestn., 2004, vol. 56, pp. 91–104.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Kurbanov, V.M., Dependence of the rate of equiconvergence on the module of continuity of potential of the Sturm–Liouville operator, Stud. Sci. Math. Hung., 2004, vol. 41, no. 3, pp. 347–364.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kurbanov, V.M. and Safarov, R.A., On uniform convergence of orthogonal expansions in eigenfunctions of Sturm–Liouville operator, Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 2004, vol. 24, no. 1, pp. 161–167.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Samarskaya, T.A., Absolute and uniform convergence of expansions in eigenfunctions and associated functions of a nonlocal boundary-value problem of the first kind, Differ. Equations, 1989, vol. 25, no. 7, pp. 813–817.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Lomov, I.S., Uniform convergence of biorthogonal series for the Schrödinger operator with multipoint boundary conditions, Differ. Equations, 2002, vol. 38, no. 7, pp. 941–948.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kurbanov, V.M. and Safarov, R.A., Uniform convergence of biorthogonal expansion corresponding to the Schrödinger operator, Proc. Inst. Math. Mech. Nat. Acad. Sci. Azerb., 2004, vol. 20, pp. 63–70.Google Scholar
  9. 9.
    Kurbanov, V.M. and Ibadov, E.D., On the properties of systems of root functions of a second-order discontinuous operator, Dokl. Math., 2009, vol. 80, no. 1, pp. 516–520.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kornev, V.V. and Khromov, A.P., On absolute convergence of expansions in eigen-and associated functions of differential and integral operators, Dokl. Ross. Akad. Nauk, 2005, vol. 400, no. 3, pp. 304–308.Google Scholar
  11. 11.
    Kornev, V.V., Sufficient criterion for absolute and uniform convergence in eigenfunctions of differential operators, Sb. Nauch. Trudov Mat. Mekh. Saratov, 2006, no. 8, pp. 56–58.Google Scholar
  12. 12.
    Kurbanov, V.M., Conditions for the absolute and uniform convergence of the biorthogonal series corresponding to a differential operator, Dokl. Math., 2008, vol. 78, no. 2, pp. 748–750.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bari, N.K., Trigonometricheskie ryady (Trigonometric Series), Moscow: Gos. Izd. Fiz. Mat. Lit., 1961.Google Scholar
  14. 14.
    Il’in, V.A., Spektral’naya teoriya differentsial’nykh operatorov (Spectral Theory of Differential Operators), Moscow: Nauka, 1991.zbMATHGoogle Scholar
  15. 15.
    Kurbanov, V.M., Equiconvergence of biorthogonal expansions in root functions of differential operators. II, Differ. Equations, 2000, vol. 36, no. 3, pp. 319–335.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kerimov, N.B., Some properties of the eigenfunctions and associated functions of ordinary differential operators, Sov. Math. Dokl., 1987, vol. 34, pp. 573–575.zbMATHGoogle Scholar
  17. 17.
    Kurbanov, V.M., On an analog of the Riesz theorem and the basis property of the system of root functions of a differential operator in Lp: II, Differ. Equations, 2013, vol. 49, no. 4, pp. 437–449.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kurbanov, V.M., On the eigenvalue distribution and a Bessel property criterion for root functions of a differential operator: I, Differ. Equations, 2005, vol. 41, no. 4, pp. 489–505.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Il’in, V.A., On the unconditional basis property for systems of eigenfunctions and associated functions of a second-order differential operator on a closed interval, Sov. Math. Dokl., 1983, vol. 28, pp. 443–447.Google Scholar
  20. 20.
    Zygmund, A, Trigonometric Series, Vol. 2, Cambridge: Cambridge Univ. Press, 1960. Translated under the title Trigonometricheskie ryady, Vol. 2, Moscow: Mir, 1965.zbMATHGoogle Scholar
  21. 21.
    Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, New York: McGraw-Hill, 1955. Translated under the title Teoriya obyknovennykh differentsial’nykh uravnenii, Moscow: Inostrannaya Literatura, 1958.Google Scholar
  22. 22.
    Kamke, E., Differentialgleichungen, Berlin: 1943. Translated under the title Spravochnik po obyknovennym differentsial’nym uravneniyam, Moscow: Inostrannaya Literatura, 1958; Nauka, 1976.zbMATHGoogle Scholar
  23. 23.
    Janczevsky, S.A., Oscillation theorems for the differential boundary value problems of the fourth order, Ann. Math., 1928, vol. 29, no. 2, pp. 521–542.MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of Mathematics and MechanicsNational Academy of Sciences of AzerbaijanBakuAzerbaijan
  2. 2.N. Tusi Azerbaijan State Pedagogical UniversityBakuAzerbaijan

Personalised recommendations