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Bases in Hilbert Spaces

  • Bao-Zhu GuoEmail author
  • Jun-Min Wang
Chapter
  • 268 Downloads
Part of the Communications and Control Engineering book series (CCE)

Abstract

This chapter tries to collect some basic facts for Riesz basis with simple introduction nature. These facts are minimal for understanding the applications in later chapters. These include basis property of exponential family \(e^{i\lambda _{n_{t}}}\), which is also referred to as nonharmonic Fourier series; the property of entire functions, specially for entire function of exponential type; the materials about the Hardy space and Blaschke are only limited to a few results; The D-type operator and One-rank perturbation for D-type operator are also introduced. The materials of generalized divided difference (GDD) developed by Russian school is also introduced. The Pavlov theorem and Keldysh theorem are specially introduced. It also presents the general results on the Riesz basis property for \(C_0\)-groups and semigroups

References

  1. Arendt W, Grabosch A, Greiner G, Groh U, Lotz HP, Moustakas U, Nagel R, Neubrander F, Schlotterbeck U (1986) One-parameter semigroups of positive operators, Lecture notes in mathematics, vol 1184. Springer, BerlinGoogle Scholar
  2. Avdonin SA, Ivanov SA (2002) Riesz bases of exponentials and divided differences. St. Petersburg Math J 13:339–351MathSciNetzbMATHGoogle Scholar
  3. Avdonin S, Moran W (2001) Ingham-type inequalities and Riesz bases of divided differences. Int J Appl Math Comput Sci 11:803–820MathSciNetzbMATHGoogle Scholar
  4. Avodonin SA, Ivanov SA (1995) Families of exponentials. Cambridge University Press, CambridgeGoogle Scholar
  5. Dunford N, Schwartz JT (1971) Linear operators, Part III. Wiley-Interscience, New YorkzbMATHGoogle Scholar
  6. Gohberg I, Goldberg S, Kaashoek M (1990) Classes of linear operators, vol 49. Vol. I, Operator theory: advances and applications. Birkhäuser, BaselCrossRefGoogle Scholar
  7. Guo BZ, Luo YH (2002) Riesz basis property of a second order hyperbolic system with collocated scalar input/output. IEEE Trans Autom Control 47:693–698MathSciNetCrossRefGoogle Scholar
  8. Guo BZ, Xu GQ (2006) Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition. J Funct Anal 231:245–268MathSciNetCrossRefGoogle Scholar
  9. Hardy GH, Littlewood DE, Polya G (1952) Inequalities. Cambridge University Press, CambridgezbMATHGoogle Scholar
  10. Hruščëv SV, Nikol’skiǐ NK, Pavlov BS (1981) Unconditional bases of exponentials and of reproducing kernels. Complex analysis and spectral theory. Lecture notes in mathematics, vol 864. Springer, Berlin, pp 214–335Google Scholar
  11. John B (1981) Bounded analytic functions. Academic Press, New YorkzbMATHGoogle Scholar
  12. Komornik V, Loreti P (2005) Fourier series in control theory. Spinger Science+Business Media Inc, New YorkCrossRefGoogle Scholar
  13. Koosis P (1998) Introduction to \(H_p\) spaces, 2ed edn. Cambridge University Press, CambridgezbMATHGoogle Scholar
  14. Levin BJa (1980) Distribution of zeros of entire functions, Revised version. AMS, ProvidenceGoogle Scholar
  15. Li BR (1978) The perturbation theory of a class of linear operators with applications. Acta Math Sinica 21(3):206–222 (in Chinese)MathSciNetzbMATHGoogle Scholar
  16. Nikol’skiĭ NK (1986) Treatise on the shift operator: spectral function theory. Springer, BerlinCrossRefGoogle Scholar
  17. Pavlov BS (1979) Basicity of exponential system and Muckenhoupt’s condition. Soviet Math Dokl 20:655–659zbMATHGoogle Scholar
  18. Rao BP (1997) Optimal energy decay rate in a damped Rayleigh beam. Contemp Math 209:221–229. RI, ProvidenceGoogle Scholar
  19. Shkalikov AA (1986) Boundary problems for ordinary differential equations with parameter in the boundary conditions. J Soviet Math 33:1311–1342CrossRefGoogle Scholar
  20. Singer I (1970) Bases in banach spaces I. Springer, BerlinCrossRefGoogle Scholar
  21. Titchmarsh EG (1953) The theory of functions. Oxford University Press, OxfordGoogle Scholar
  22. Xu GQ, Guo BZ (2003) Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation. SIAM J Control Optim 42:966–984MathSciNetCrossRefGoogle Scholar
  23. Young RM (2001) An introduction to nonharmonic fourier series, Revised first edition edn. Academic Press, LondonzbMATHGoogle Scholar
  24. Zwart H (2010) Riesz basis for strongly continuous groups. J Differ Equ 249:2397–2408MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems ScienceAcademia SinicaBeijingChina
  2. 2.School of Computer Science and Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa
  3. 3.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina

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