Periodica Mathematica Hungarica

, Volume 22, Issue 3, pp 187–196 | Cite as

On some riesz bases



In this paper we consider three problems concerning systems of vector exponentials. In the first part we prove a conjecture of V. Komornik raised in [14] on the independence of the movement of a rectangular membrane in different points. It was independently proved by M. Horváth [9] and S. A. Avdonin (personal communication). The analogous problem for the circular membrane was partly solved in [3] — the complete solution is given in [10]. In the second part we fill in a gap in the theory of Blaschke-Potapov products developed in the paper [19] of Potapov. Namely we prove that the Blaschke-Potapov product is determined by its kernel sets up to a multiplicative constant matrix. In the third part of the present paper we give a multidimensional generalization of the notion of sine type function developed by Levin [16], [17] and by our generalization we prove the multidimensional variant of the Levin-Golovin basis theorem [16], [6].

Mathematics subject classification numbers, 1980/1985



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  1. [1]
    S. A.AVDONIN, On Riesz bases from exponential functions inL 2 in Russian Vestnik Leningradskovo Gos. Univ. Ser. Mat.13 (1974), 5–12.Google Scholar
  2. [2]
    S. A.AVDONIN, S. A.IVANOV and I.JOÓ, On Riesz bases from vector exponentials I, II, Annales Univ. Sci. Budapest. Sect. Math.,32 (1989), 101–114, 115–126.Google Scholar
  3. [3]
    A.BOGMÉR, M.HORVÁTH and I.JOÓ, Notes to some papers of V. Komornik on vibrating membranes, Periodica Math. Hung.,20 (3) (1989), 193–205.Google Scholar
  4. [4]
    J. W. S. CASSELS, An introduction to Diophantine approximation, Cambridge Univ. Press, 1957.MR 19, 396Google Scholar
  5. [5]
    R. J.DUFFIN and J. J.EACHUS, Some notes on an expansion theorem of Paley and Wiener, Bull. Am. Math. Soc.48 (1942), 850–855.Google Scholar
  6. [6]
    V. A.GOLOVIN, On the stability of exponential bases (in Russian), Dokl. A. N. Arm. SSR36 (2) (1963), 65–70.Google Scholar
  7. [7]
    K.HOFFMAN, Banach spaces of analytic functions, Prentice Hall, Englewood Cliffs, New Yersey, 1962.Google Scholar
  8. [8]
    M.HORVÁTH, Vibrating strings with free ends, Acta Math. Hung.51 (1–2) (1988), 171–180.Google Scholar
  9. [9]
    M. HORVÁTH, The vibration of a membrane in different points, Annales Univ. Sci. Budapest., Sect. Math. (to appear).Google Scholar
  10. [10]
    I. JOÓ, A remark on the vibration of a circular membrane in different points, Acta Math. Hung. (to appear).Google Scholar
  11. [11]
    I.JOÓ, On the rechability set of a string, Acta Math. Hung.49 (1–2) (1987), 203–211Google Scholar
  12. [12]
    I. JOÓ, Contrôlabilité exact propriétés d'oscillation de l'équation des ondes par anaylse nonharmonique, C. R. Acad. Sci. Paris, t.312, Serie 1, (1991) 119–122.Google Scholar
  13. [13]
    V. E.KATZNELSON, On bases from exponential functions inL 2 (in Russian) Funk. Anal. i jevo Priloz.5 (1) (1971), 37–47.Google Scholar
  14. [14]
    V.KOMORNIK, On the vibrations of a square membrane, Proc. Soc. Edinburgh111A (1989), 13–20.Google Scholar
  15. [15]
    B. Ya.LEVIN, Distribution of zeros of entire functions (in Russian), GITTL, Moscow 1956.Google Scholar
  16. [16]
    B. Ja.LEVIN, On the bases from exponential functions inL 2 (in Russian), Zapiski Mat. otgyel. fiz.-mat. fakulteta Harkovskovo univ.27 (4) (1961), 39–48.Google Scholar
  17. [17]
    B. Ja.LEVIN, I. V.OSTROVSKII, Small perturbations of the set of roots of sine-type functions (in Russian), Izvestija Akad. Nauk SSSR, Ser. Mat.43 (1) (1979), 87–110.MR 82b: 30022Google Scholar
  18. [18]
    N. K.NIKOLSKII, Lectures on the shift operator (in Russian), Nauka, Moscow 1980.MR 82i: 47013Google Scholar
  19. [19]
    V. P. POTAPOV, The multiplicative structure ofJ-contractive matrix-functions (in Russian), Trudi Moskovskovo Mat. Obsestva No4 (1955), 125–236.MR 17, 958Google Scholar

Copyright information

© Akadémiai Kiadó 1991

Authors and Affiliations

  • I. Joó
    • 1
  1. 1.Mathematical Institute of theHungarian Academy of SciencesBudapestHungary

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