Skip to main content
SpringerLink
Log in

Polynomials in the de Branges spaces of entire functions

  • Published:
Arkiv för Matematik

Abstract

We study the problem of density of polynomials in the de Branges spaces ℋ(E) of entire functions and obtain conditions (in terms of the distribution of the zeros of the generating function E) ensuring that the polynomials belong to the space ℋ(E) or are dense in this space. We discuss the relation of these results with the recent paper of V. P. Havin and J. Mashreghi on majorants for the shift-coinvariant subspaces. Also, it is shown that the density of polynomials implies the hypercyclicity of translation operators in ℋ(E).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Bibliography

  1. Akhiezer, N. I., On a generalization of the Fourier transformation and the Paley–Wiener theorem, Dokl. Akad. Nauk SSSR96 (1954), 889–892 (Russian).

    Google Scholar 

  2. Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). English transl.: Hafner, New York, 1965.

  3. Baranov, A. D., Isometric embeddings of the spaces KΘ in the upper half-plane, in Function Theory and Partial Differential Equations, Problems in Math. Anal. 21, pp. 30–44, St. Petersbg. Gos. Univ., St. Petersburg, 2000 (Russian). English transl.: J. Math. Sci. (New York ) 105 (2001), 2319–2329.

  4. Baranov, A. D., The Bernstein inequality in the de Branges spaces and embedding theorems, Proc. St. Petersburg Math. Soc.9 (2001), 23–53 (Russian). English transl.: Amer. Math. Soc. Transl.209 (2003), 21–49.

  5. Baranov, A. D., On estimates of the Lp-norms of the derivatives in spaces of entire functions, Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 303 (2003), 5–33, 321 (Russian). English transl.: J. Math. Sci. (N. Y.) 129 (2005), 3927–3943.

  6. Baranov, A. D. and Havin, V. P., Admissible majorants for model subspaces, and arguments of inner functions, Funktsional. Anal. i Prilozhen.40 (2006), to appear.

  7. Baranov, A. D. and Woracek, H., Admissible majorants for de Branges spaces of entire functions, Preprint.

  8. Birkhoff, G. D., Démonstration d’un théoreme elementaire sur les fonctions entiéres, C. R. Acad. Sci. Paris189 (1929), 473–475.

    Google Scholar 

  9. Borichev, A. and Sodin, M., Weighted polynomial approximation and the Hamburger moment problem, in Complex Analysis and Differential Equations (Uppsala, 1997), pp.110–122, Uppsala Univ., Uppsala, 1998.

  10. Borichev, A. and Sodin, M., The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line, J. Anal. Math.76 (1998), 219–264.

    Google Scholar 

  11. de Branges, L., Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, NJ, 1968.

  12. Chan, K. C. and Shapiro, J. H., The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J.40 (1991), 1421–1449.

    Google Scholar 

  13. Godefroy, G. and Shapiro, J. H., Operators with dense, invariant cyclic vector manifolds, J. Funct. Anal.98 (1991), 229–269.

    Google Scholar 

  14. Gurarii, V. P., The Fourier transform in L2(-∞,∞) with weight, Mat. Sb.58 (1962), 439–452 (Russian).

    Google Scholar 

  15. Havin, V. P. and Mashreghi, J., Admissible majorants for model subspaces of H2. Part I: Slow winding of the generating inner function, Canad. J. Math.55 (2003), 1231–1263.

    Google Scholar 

  16. Havin, V. P. and Mashreghi, J., Admissible majorants for model subspaces of H2. Part II: Fast winding of the generating inner function, Canad. J. Math.55 (2003), 1264–1301.

    Google Scholar 

  17. Kaltenbäck, M. and Woracek, H., Hermite–Biehler functions with zeros close to the imaginary axis, Proc. Amer. Math. Soc.133 (2005), 245–255.

    Google Scholar 

  18. Koosis, P., Measures orthogonales extrémales pour l’approximation pondérée par des polynômes, C. R. Acad. Sci. Paris Sér. I Math.311 (1990), 503–506.

  19. Levin, B. Ya., Distribution of Zeros of Entire Functions, GITTL, Moscow, 1956 (Russian). English transl.: Amer. Math. Soc., Providence, RI, 1964; rev. edn.: Amer. Math. Soc., Providence, RI, 1980.

  20. MacLane, G. R., Sequences of derivatives and normal families, J. Anal. Math.2 (1952), 72–87.

    Google Scholar 

  21. Nikolski, N. K. Treatise on the Shift Operator. Special Function Theory, Springer, Berlin, 1986.

  22. Nikolski, N. K., Operators, Functions, and Systems: an Easy Reading, Vol. 1–2, Mathematical Surveys and Monographs 92–93, Amer. Math. Soc., Providence, RI, 2002.

  23. Woracek, H., De Branges spaces of entire functions closed under forming difference quotients, Integral Equations Oper. Theory37 (2000), 238–249.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Mechanics, St. Petersburg State University, 28, Universitetskii pr., St. Petersburg, 198504, Russia

    Anton D. Baranov

  2. Laboratoire d’Analyse et Géométrie, Université Bordeaux 1, 351, Cours de la Libération, FR-33405, Talence, France

    Anton D. Baranov

Authors
  1. Anton D. Baranov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anton D. Baranov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baranov, A. Polynomials in the de Branges spaces of entire functions. Ark Mat 44, 16–38 (2006). https://doi.org/10.1007/s11512-005-0006-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11512-005-0006-1

Keywords

Search

Navigation