Mathematical Notes

, Volume 61, Issue 3, pp 301–312

Entire functions of bernstein's class that are not fourier-stieltjes transforms

  • A. M. Sedletskii


We consider certain subclasses of the class of entire functions of exponential type bounded on the real axis. We construct functions that belong to these subclasses but are not Fourier-Stieltjes transforms. Particular attention is given to the distribution of zeros of such functions. The results obtained allow us to study the stability of completeness of systems of exponentials inC andLp under small perturbations of the exponents.

Key words

entire functions of exponential type Bernstein's class distribution of zeros 


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  1. 1.
    N. I. Akhiezer,Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).Google Scholar
  2. 2.
    R. P. Boas,Entire Functions, Acad. Press, New York (1954).Google Scholar
  3. 3.
    B. Ya. Levin, “Interpolation by entire functions of exponential type,” in:Mathematical Physics and Functional Analysis [in Russian], Vol. 1, Kharkov Institute of Physics and Engineering, Ukrainian Academy of Sciences, Kharkov (1969), pp. 136–146.Google Scholar
  4. 4.
    A. M. Sedletskii, “Biorthogonal expansions in series in exponential functions,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],36, No. 3, 583–590 (1972).MathSciNetGoogle Scholar
  5. 5.
    B. Ya. Levin and Yu. I. Lyubarskii, “Interpolation by entire functions from special classes and related expansions in exponential series,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],39, No. 3, 657–702 (1975).Google Scholar
  6. 6.
    A. M. Sedleckil, “On completeness of the systems {exp(ix(n+ih n))},”Anal. Math.,2, No. 2, 125–143 (1978).Google Scholar
  7. 7.
    A. M. Sedletskii, “Biorthogonal expansions of functions into exponential series on intervals of the real line,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],37, No. 5, 51–95 (1982).MATHMathSciNetGoogle Scholar
  8. 8.
    E. Seneta,Regularly Varying Functions, Springer-Verlag, Berlin (1976).Google Scholar
  9. 9.
    A. M. Sedletskii, “On uniform convergence of nonharmonic Fourier series,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],200, 299–309 (1991).MATHGoogle Scholar
  10. 10.
    R. Edwards,Fourier Series. A Modern Introduction, Vol. 2, Springer, Heidelberg (1982).Google Scholar
  11. 11.
    A. Zygmund,Trigonometric Series, Vol. 1, Cambridge Univ. Press, Cambridge (1959).Google Scholar
  12. 12.
    P. Koosis,Introduction to H p Spaces, Cambridge Univ. Press, Cambridge (1980).Google Scholar
  13. 13.
    G. Polya, “Über die Nullstellen gewisser ganzer Funktionen,”Math. Z.,2, 352–383 (1918).MATHMathSciNetGoogle Scholar
  14. 14.
    W. O. Alexander and R. Redheffer, “The excess of sets of complex exponentials,”Duke Math. J.,34, No. 1, 59–72 (1967).MathSciNetGoogle Scholar
  15. 15.
    A. M. Sedletskii, “On stability of completeness and minimality of a system of exponentials inL 2,”Mat. Zametki [Math. Notes],15, No. 2, 213–219 (1974).Google Scholar
  16. 16.
    A. M. Sedletskii, “Excesses of close systems of exponentials inL p,”Sibirsk. Mat. Zh. [Siberian Math. J.],24, No. 4, 164–175 (1983).MATHMathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • A. M. Sedletskii
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityRussia

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