Mathematical Notes

, Volume 62, Issue 3, pp 350–355 | Cite as

A first-order boundary value problem with boundary condition on a countable set of points

  • A. M. Minkin
Article
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Abstract

LetE={E n } be the family of subspaces spanning the eigenfunctions and adjoint functions of the boundary-value problem
$$ - i\frac{{dy}}{{dx}} = \lambda y, - \alpha \leqslant x \leqslant \alpha , U(y) \equiv \int_{ - a}^a {y(t)} d\sigma (t) = 0$$
that correspond to “close” eigenvalues (in the sense of the distance defined as the maximal of the Euclidean and the hyperbolic metrics). For a purely discrete measuredσ, it is shown that the systemE does not form an unconditional basis of subspaces inL2(−a, a) if at least one of the end points ±a is mass-free.

Key Words

Boundary value problem eigenfunctions adjoint functions discrete measure unconditional basis mass-free end points 

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References

  1. 1.
    B. Ya. Levin, “Bases of exponentials inL 2,”Zapiski Matem. Otd. Fiz.-Matem. Fakul’teta Khar’kovskogo Univ. i Khar’kovskogo Matem. Obshchestva,27, No. 4, 39–48 (1961).Google Scholar
  2. 2.
    V. D. Golovin, “Biorthogonal expansions in linear combinations of exponentials inL 2,”Zapiski Matem. Otd. Fiz.-Matem. Fakul’teta Khar’kovskogo Univ. i Khar’kovskogo Univ. i Khar’kovskogo Matem. Obshchestva,30, No. 4, 18–29 (1963).Google Scholar
  3. 3.
    B. S. Pavlov, “Spectral analysis of differential operators with a “diffuse” boundary condition,” in:Problems of Mathematical Physics [in Russian], No. 6, Izd. Leningr. Univ., Leningrad (1973), pp. 101–109.Google Scholar
  4. 4.
    V. A. Molodenkov, “Expansion in eigenfunctions of a boundary value problem,” in:Differential Equations and Computational Mathematics [in Russian], No. 2, Saratov (1975), pp. 56–65.Google Scholar
  5. 5.
    V. A. Molodenkov and A. P. Khromov, “Expansion in eigenfunctions of a boundary value problem for the differentiation operator,” in:Differential Equations and Computational Mathematics [in Russian], No. 1, Saratov (1972), pp. 17–26.Google Scholar
  6. 6.
    A. M. Sedletskii, “Biorthogonal expansions of functions in series of exponentials in a finite interval on the real axis,”Uspekhi Mat. Nauk [Russian Math. Surveys],37, No. 5, (227), 51–95 (1982).MATHMathSciNetGoogle Scholar
  7. 7.
    A. M. Sedletskii, “Expansion in eigenfunctions of the differentiation operator with a diffuse boundary condition,”Differentsial’nye Uravneniya [Differential Equations],30, No. 1, 70–76 (1994).MATHMathSciNetGoogle Scholar
  8. 8.
    A. M. Sedletskii, “Uniform convergence of nonharmonic Fourier series,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],200, 299–309 (1991).MATHGoogle Scholar
  9. 9.
    B. Ya. Levin,Distribution of the Roots of Entire Functions [in Russian], Gostekhizdat, Moscow (1956).Google Scholar
  10. 10.
    G. M. Gubreev, “Spectral analysis of biorthogonal expansions of functions in series with exponentials,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],53, No. 6, 1236–1268 (1989).Google Scholar
  11. 11.
    B. S. Pavlov, “Basis Property of a System of Exponentials and the Muckenhoupt condition,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],247, No. 1, 37–40 (1979).MATHMathSciNetGoogle Scholar
  12. 12.
    N. K. Nikol’skii, “Bases of exponentials and reproducing kernels,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],252, No. 6, 1316–1320 (1980).MathSciNetGoogle Scholar
  13. 13.
    S. V. Khrushchev, “Perturbation theorems for bases of exponentials and the Muchenhoupt condition,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],247, No. 1, 44–48 (1979).MathSciNetGoogle Scholar
  14. 14.
    B. S. Pavlov, N. K. Nikol’skii [Nikolskii], and S. V. Khrushchev [Hrushev], “Unconditional bases of exponentials and reproducing kernels,”Lecture Notes in Math.,864, 214–335 (1981).Google Scholar
  15. 15.
    A. M. Minkin, “Reflection of exponents and unconditional bases of exponentials,”Algebra i Analiz,3, No. 5, 110–135 (1991).MathSciNetGoogle Scholar
  16. 16.
    S. A. Avdonin and I. Joó, “Riesz bases of exponentials and sine-type functions,”Acta Math. Hungar.,51, No. 1–2, 3–14 (1988).MathSciNetGoogle Scholar
  17. 17.
    S. A. Avdonin, I. Joó, and M. Horváth, “Riesz bases of elements of the form\(x^k e^{i\lambda _n x} \)Vestnik Leningrad. Univ. Mat. Mekh. Astronom. [Vestnik Leningrad Univ. Math.], No. 4 (22), 3–7 (1989).Google Scholar
  18. 18.
    G. M. Gubreev and T. R. Ignatenko, “A class of orthogonalizers of the families of exponentials with real frequencies,”Ukrain. Mat. Zh. [Ukrainian Math. J.]44, No. 8, 1031–1044 (1992).MathSciNetGoogle Scholar
  19. 19.
    A. E. Eremenko and M. L. Sodin, “Parametrization of entire functions of sine type by their critical values,”Adv. Soviet Math.,11, 237–242 (1992).MathSciNetGoogle Scholar
  20. 20.
    A. M. Minkin, “Projection method and unconditional bases,” in:Algebraic and Topological Methods in Mathematical Physics [in Russian], Katsiveli (Ukrain) (1994).Google Scholar
  21. 21.
    N. K. Nikol’skii,Lectures on Translation Operators [in Russian], Nauka, Moscow (1980).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • A. M. Minkin
    • 1
  1. 1.Saratov State UniversitySaratov USUSSR

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