Control of Wave and Beam PDEs pp 27-195 | Cite as

# Bases in Hilbert Spaces

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## 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

- 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
- Avdonin SA, Ivanov SA (2002) Riesz bases of exponentials and divided differences. St. Petersburg Math J 13:339–351MathSciNetzbMATHGoogle Scholar
- Avdonin S, Moran W (2001) Ingham-type inequalities and Riesz bases of divided differences. Int J Appl Math Comput Sci 11:803–820MathSciNetzbMATHGoogle Scholar
- Avodonin SA, Ivanov SA (1995) Families of exponentials. Cambridge University Press, CambridgeGoogle Scholar
- Dunford N, Schwartz JT (1971) Linear operators, Part III. Wiley-Interscience, New YorkzbMATHGoogle Scholar
- Gohberg I, Goldberg S, Kaashoek M (1990) Classes of linear operators, vol 49. Vol. I, Operator theory: advances and applications. Birkhäuser, BaselCrossRefGoogle Scholar
- 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
- 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
- Hardy GH, Littlewood DE, Polya G (1952) Inequalities. Cambridge University Press, CambridgezbMATHGoogle Scholar
- 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
- John B (1981) Bounded analytic functions. Academic Press, New YorkzbMATHGoogle Scholar
- Komornik V, Loreti P (2005) Fourier series in control theory. Spinger Science+Business Media Inc, New YorkCrossRefGoogle Scholar
- Koosis P (1998) Introduction to \(H_p\) spaces, 2ed edn. Cambridge University Press, CambridgezbMATHGoogle Scholar
- Levin BJa (1980) Distribution of zeros of entire functions, Revised version. AMS, ProvidenceGoogle Scholar
- 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
- Nikol’skiĭ NK (1986) Treatise on the shift operator: spectral function theory. Springer, BerlinCrossRefGoogle Scholar
- Pavlov BS (1979) Basicity of exponential system and Muckenhoupt’s condition. Soviet Math Dokl 20:655–659zbMATHGoogle Scholar
- Rao BP (1997) Optimal energy decay rate in a damped Rayleigh beam. Contemp Math 209:221–229. RI, ProvidenceGoogle Scholar
- Shkalikov AA (1986) Boundary problems for ordinary differential equations with parameter in the boundary conditions. J Soviet Math 33:1311–1342CrossRefGoogle Scholar
- Singer I (1970) Bases in banach spaces I. Springer, BerlinCrossRefGoogle Scholar
- Titchmarsh EG (1953) The theory of functions. Oxford University Press, OxfordGoogle Scholar
- 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
- Young RM (2001) An introduction to nonharmonic fourier series, Revised first edition edn. Academic Press, LondonzbMATHGoogle Scholar
- Zwart H (2010) Riesz basis for strongly continuous groups. J Differ Equ 249:2397–2408MathSciNetCrossRefGoogle Scholar