Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T07:24:42.975Z Has data issue: false hasContentIssue false
  • English
  • Français

On Asymptotically OrthonormalSequences

Published online by Cambridge University Press:  20 November 2018

Emmanuel Fricain  and
Rishika Rupam
Emmanuel Fricain
Affiliation:
Laboratoire Paul Painlevé, Université Lille 1, 59 655 Villeneuve d'Ascq Cédex e-mail: [email protected], [email protected]
Rishika Rupam
Affiliation:
Laboratoire Paul Painlevé, Université Lille 1, 59 655 Villeneuve d'Ascq Cédex e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An asymptotically orthonormal sequence is a sequence that is nearly orthonormal in the sense that it satisfies the Parseval equality up to two constants close to one. In this paper, we explore such sequences formed by normalized reproducing kernels for model spaces and de Branges–Rovnyak spaces.

Keywords

30J0530H1046E22function spacede Branges-Rovnyak and model spacereproducing kernelasymptotically orthonormal sequence
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 6 , 01 December 2017 , pp. 1312 - 1337
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Baranov, A. D., Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings. J. Funct. Anal. 223(2005), no. 1,116–146. http://dx.doi.Org/10.1016/j.jfa.2OO4.08.014 Google Scholar
[2] Baranov, A. D., Stability of bases and frames of reproducing kernels in model spaces. Annales de l'Institut 2399–2422. http://dx.doi.org/10.5802/aif.2165 Google Scholar
[3] Baranov, A., Fricain, E., and Mashreghi, J., Weighted norm inequalities for de Branges-Rovnyak spaces and their applications. Amer. J. Math. 132(2010), no. 1,125–155.Google Scholar
[4] Chalendar, I., Fricain, E., and Timotin, D., Functional models and asymptotically orthonormal sequences. Annales de l'Institut Fourier 53(2003), no. 5,1527–1549. http://dx.doi.Org/10.58O2/aif.1987 Google Scholar
[5] Cohn, W. S., Carleson measures and operators on sta-invariant subspaces. J. Operator Theory 15(1986), no. 1, 181–202.Google Scholar
[6] de Branges, L. and Rovnyak, J., Canonical models in quantum scattering theory. In: Perturbation theory and its Applications in Quantum Mechanics, 1966, pp. 295–392.Google Scholar
[7] Rovnyak, J.,, Square summable power series. Courier Corporation, 2015.Google Scholar
[8] Fricain, E. and Mashreghi, J., Boundary behavior of functions in the de Branges-Rovnyak spaces. Complex Analysis Operator Theory 2(2008), no. 1, 87–97.http://dx.doi.org/10.1007/s11785-007-0028-8 Google Scholar
[9] Fricain, E., The theory ofℋ(b) spaces. Vol. 1, New Mathematical Monographs, 20, Cambridge University Press, Cambridge, 2016.http://dx.doi.Org/10.1017/CBO9781139226752 Google Scholar
[10] Garcia, S. R., Mashreghi, J., and Ross, W. T., Introduction to model spaces and their operators. Cambridge Studies in Advanced Mathematics, 148, Cambridge University Press, Cambridge, 2016.http://dx.doi.org/10.1017/CBO9781316258231 Google Scholar
[11] Gorkin, P., J. McCarthy, E., Pott, S., and Wick, B. D., Thin sequences and the Gram matrix. Arch. Math. 103(2014), 93–99.http://dx.doi.Org/10.1007/s00013-014-0667-8 Google Scholar
[12] Hruschev, S. V., Nikolskii, N. K., and Pavlov, B.S., Unconditional hases of exponentials and of reproducing kernels. In: Complex analysis and spectral theory (Leningrad, 1979/1980), Lecture Notes in Math., 864, Springer, Berlin-New York, 1981, pp. 214–335.Google Scholar
[13] Levinson, N., Gap and density theorems. American Mathematical Society Colloquium Publications, 26, American Mathematical Society, New York, 1940.Google Scholar
[14] Nikolski, N. K., Treatise on the shift operator-spectral function theory. Grundlehren der Mathematischen Wissenschaften, 273, Springer-Verlag, Berlin-Heidelberg, 1986.http://dx.doi.org/10.1007/978-3-642-70151-1 Google Scholar
[15] Nikolski, N. K., Operators, functions, and systems - an easy reading: Hardy, Hankel, and Toeplitz. American Mathematical Society, Providence, RI, 2009.Google Scholar
[16] R. P. and Wiener, N., Fourier transforms in the complex domain. American Mathematical Society Colloquium Publications, 19, American Mathematical Society, Providence, RI, 1987.Google Scholar
[17] Sarason, D., Sub-Hardy Hilbert spaces in the unit disk. University of Arkansas Lecture Notes in the Mathematical Sciences, 10, John Wiley & Sons, Inc., New York, 1994.Google Scholar
[18] Vinogradov, S. A. and Havin, V. P., Free interpolation in H∞ and in certain other classes of functions. I. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 47(1974), 15-54, 184–185,191.Google Scholar
[19] Volberg, A., Two remarks concerning the theorem ofS. Axler, S.-Y.A. Chang, and D. Sarason. J. Operator Theory 8(1982), 209–218.Google Scholar
[20] Young, R. M., An introduction to nonharmonic Fourier series. First ed., Academic Press, Inc., San Diego, CA, 2001.Google Scholar