Russian Mathematics

, Volume 63, Issue 2, pp 51–61 | Cite as

Existence of Frames Based on the Szegö Kernel in the Hardy Space

  • K. S. SperanskyEmail author
  • P. A. TerekhinEmail author


As is known, a sequence of functions consisting of meanings of the Szegö reproducing kernel of the Hardy space in the unit disk cannot be Duffin—Schaeffer frame. In the present paper we show that the using of more general conception of frame enables us to solve positively the question on existence of a generalized frame built in terms of the Szegö kernel.

Key words

Duffin—Schaeffer frame Banach frame model space of frame reproducing kernel of Hilbert space Hardy space Szegö kernel 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The research is performed with support of Russian Foundation for Basic Researches (grant No. 18-01-00414).


  1. 1.
    Lukashenko, T. P. “On Properties of Ortho-Recursive Expansions in Non-Orthogonal Systems”, Vestn. Moscow Univ. Ser.1.Math. Mech. 1, 6–10 (2001).Google Scholar
  2. 2.
    Galatenko, V. V., Lukashenko, T. P., Sadovnichii, V. A. On Properties of Ortho-Recursive Expansions in Subspaces, Trudy MIAN, 284, 138–141 (2014).Google Scholar
  3. 3.
    Temlyakon, V. N. “Greedy Approximation”, Acta Numer. 17, 235–409 (2008).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Duffin, R. J., Schaeffer, A. C. “A Class of Nonharmonic Fourier Series”, Trans. Amer. Math. Soc. 72, 341–366 (1952).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Feichtinger, H. G., Gröchenig, K. “Banach Spaces Related to Integrable Group Representations and Their Atomic Decomposition”. I, J. Funct. Anal. 86, 307–340 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Feichtinger, H. G., Groöchenig, K. “Banach Spaces Related to Integrable Group Representations and Their Atomic Decomposition”. II, Monatsh. Math. 108, 129–148 (1989).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gröchenig, K. “Describing Functions: Atomic Decompositions Versus Frames”, Monatsh. Math. 112, No. 1, 1–41 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Casazza, P. G., Han, D., Larson, D. R. “Frames for Banach Spaces”, Contemp. Math. 247, 149–182 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Casazza, P. G., Christensen, O., Stoeva, D. “Frame Expansion in Separable Banach Spaces”, J. Math. Anal. Appl. 307, No. 2, 710–723 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duren, P., Schuster, A. Bergman spaces (AMS, Providence, RI, 2004).CrossRefzbMATHGoogle Scholar
  11. 11.
    Bari, N. K. “Bi-Orthogonal Systems and Bases of Hilbert Space”, Mathem. IV, Uchen. Zap. Moscow Univ., 148, 69–107 (1951).Google Scholar
  12. 12.
    Christensen, O. An introduction to Frames and Riesz Bases (2nd rev. ed., Appl. Numer. Harmon. Anal., Birkhauser/Springer, New York, 2016).Google Scholar
  13. 13.
    Partington, J. R. Interpolation, Indentification, and Sampling (Clarendon Press, Oxford, 1997).Google Scholar
  14. 14.
    Zhang, H., Zhang, J. “Frames, Riesz Bases, and Sampling Expansions in Banach Spaces via Semi-Inner Products”, Appl. Comput. Harmon. Anal. 31, 1–25 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Song, M.-S., Jorgensen, P. E. T. “Reproducing Kernel Hilbert Space vs. Frame Estimates”, Math. 3, 615–625 (2015).CrossRefzbMATHGoogle Scholar
  16. 16.
    Führ, H., Gröchenig, K., Haimi, A., Klotz, A., Romero, J. L. “Density of Sampling and Interpolation in Reproducing kernel Hilbert spaces”, J. London Math. Soc. 96 (2), (2017).Google Scholar
  17. 17.
    Gao, J., Harris, C., Gunn, S. “On a Class of a Support Vector Kernels Based on Frames in Function Hilbert Spaces”, Neural Comput. 13 (9), 1975–1994 (2001).CrossRefzbMATHGoogle Scholar
  18. 18.
    Rakotomamonjy, A., Canu, S. “Frames, Reproducing Kernels, Regularization and Learning”, Journal of Machine Learning Research 6, 1485–1515 (2005).MathSciNetzbMATHGoogle Scholar
  19. 19.
    Halmosh P. Hilbert spaces in problems (Mir publishers, Moscow, 1970).zbMATHGoogle Scholar
  20. 20.
    Marcus, A. W., Spielman, D. A., Srivastava, N. “Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer problem”, Ann. Math. 182 (1), 327–350 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Akhiezer, N. I., Glazman, I. M. Theory of Linear Operators in Hilbert Space (Nauka, Moscow, 1966).zbMATHGoogle Scholar
  22. 22.
    Totik, V. “Recovery of H p-Functions”, Proc. Amer. Math. Soc. 90, No. 4, 531–537 (1984).MathSciNetzbMATHGoogle Scholar
  23. 23.
    Terekhin, P. A. “Systems of Representation and Projections of Bases”, Matem. Zametki 75, No. 6, 944–947 (2004).CrossRefzbMATHGoogle Scholar
  24. 24.
    Terekhin, P. A. “Banach Frames in Problem of Affine Synthesis”, Matem. Sb. 200, No. 9, 127–146 (2009).CrossRefzbMATHGoogle Scholar
  25. 25.
    Terekhin, P. A. “Frames in Banach Space”, Func. Analys i Ego Prilozh. 44 (3), 50–62 (2010).MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schaefer, H. H., Wolff, M. P. Topological Vector Spaces (2nd ed., Grad. Text in Math., Springer-Verlag, New York, 1999).CrossRefzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Saratov State UniversitySaratovRussia

Personalised recommendations