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Статья
2019

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


K. S. SperanskyK. S. Speransky, P. A. TerekhinP. A. Terekhin
Русская математика
https://doi.org/10.3103/S1066369X19020075
Abstract / Full Text

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.

Author information
  • Saratov State University, ul. Astrakhanskaya 83, Saratov, 410012, RussiaK. S. Speransky & P. A. Terekhin
References
  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).
  2. Galatenko, V. V., Lukashenko, T. P., Sadovnichii, V. A. On Properties of Ortho-Recursive Expansions in Subspaces, Trudy MIAN, 284, 138–141 (2014).
  3. Temlyakon, V. N. “Greedy Approximation”, Acta Numer. 17, 235–409 (2008).
  4. Duffin, R. J., Schaeffer, A. C. “A Class of Nonharmonic Fourier Series”, Trans. Amer. Math. Soc. 72, 341–366 (1952).
  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).
  6. Feichtinger, H. G., Groöchenig, K. “Banach Spaces Related to Integrable Group Representations and Their Atomic Decomposition”. II, Monatsh. Math. 108, 129–148 (1989).
  7. Gröchenig, K. “Describing Functions: Atomic Decompositions Versus Frames”, Monatsh. Math. 112, No. 1, 1–41 (1991).
  8. Casazza, P. G., Han, D., Larson, D. R. “Frames for Banach Spaces”, Contemp. Math. 247, 149–182 (1999).
  9. Casazza, P. G., Christensen, O., Stoeva, D. “Frame Expansion in Separable Banach Spaces”, J. Math. Anal. Appl. 307, No. 2, 710–723 (2005).
  10. Duren, P., Schuster, A. Bergman spaces (AMS, Providence, RI, 2004).
  11. Bari, N. K. “Bi-Orthogonal Systems and Bases of Hilbert Space”, Mathem. IV, Uchen. Zap. Moscow Univ., 148, 69–107 (1951).
  12. Christensen, O. An introduction to Frames and Riesz Bases (2nd rev. ed., Appl. Numer. Harmon. Anal., Birkhauser/Springer, New York, 2016).
  13. Partington, J. R. Interpolation, Indentification, and Sampling (Clarendon Press, Oxford, 1997).
  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).
  15. Song, M.-S., Jorgensen, P. E. T. “Reproducing Kernel Hilbert Space vs. Frame Estimates”, Math. 3, 615–625 (2015).
  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).
  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).
  18. Rakotomamonjy, A., Canu, S. “Frames, Reproducing Kernels, Regularization and Learning”, Journal of Machine Learning Research 6, 1485–1515 (2005).
  19. Halmosh P. Hilbert spaces in problems (Mir publishers, Moscow, 1970).
  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).
  21. Akhiezer, N. I., Glazman, I. M. Theory of Linear Operators in Hilbert Space (Nauka, Moscow, 1966).
  22. Totik, V. “Recovery of H p-Functions”, Proc. Amer. Math. Soc. 90, No. 4, 531–537 (1984).
  23. Terekhin, P. A. “Systems of Representation and Projections of Bases”, Matem. Zametki 75, No. 6, 944–947 (2004).
  24. Terekhin, P. A. “Banach Frames in Problem of Affine Synthesis”, Matem. Sb. 200, No. 9, 127–146 (2009).
  25. Terekhin, P. A. “Frames in Banach Space”, Func. Analys i Ego Prilozh. 44 (3), 50–62 (2010).
  26. Schaefer, H. H., Wolff, M. P. Topological Vector Spaces (2nd ed., Grad. Text in Math., Springer-Verlag, New York, 1999).