Araştırma Makalesi
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Generalized eigenvectors of linear operators and biorthogonal systems

Yıl 2022, Cilt: 5 Sayı: 2, 60 - 71, 15.06.2022
https://doi.org/10.33205/cma.1077842

Öz

The notions of a set of generalized eigenvalues and a set of generalized eigenvectors of a linear operator in Euclidean space are introduced. In addition, we provide a method to find a biorthogonal system of a subsystem of eigenvectors of some linear operators in a Hilbert space whose systems of canonical eigenvectors are over-complete. Related to our problem, we will show an example of a linear differential operator that is formally adjoint to Bessel-type differential operators. We also investigate basis properties (completeness, minimality, basicity) of the systems of generalized eigenvectors of this differential operator.

Kaynakça

  • H. Bateman, A. Erdélyi: Higher transcendental functions, Vol. 2, McGraw-Hill, New York-Toronto-London (1953).
  • Yu. M. Berezanskii: Expansions in eigenfunctions of self-adjoint operators, Transl. Math. Monogr., Vol. 17. Amer. Math. Soc., Providence, RI (1968).
  • M. Bertola, M. Gekhtman and J. Szmigielski: Cubic string boundary value problems and Cauchy biorthogonal polynomials, J. Phys. A: Math. Theor., 42 (45) (2009), 13 p.
  • C. Cesarano: A note on bi-orthogonal polynomials and functions, Fluids, 5 (3) (2020), 105.
  • N. Dunford, J. T. Schwartz: Linear operators. Spectral operators, Part III. Wiley-Interscience, New York-London-Sydney-Toronto (1971).
  • R. V. Khats’: On conditions of the completeness of some systems of Bessel functions in the space $L^2((0;1);x^{2p} dx)$, Azerb. J. Math., 11 (1) (2021), 3–10.
  • R. V. Khats’: Completeness conditions of systems of Bessel functions in weighted $L^2$-spaces in terms of entire functions, Turk. J. Math., 45 (2) (2021), 890–895.
  • R. V. Khats’: Integral representation of one class of entire functions, Armen. J. Math., 14 (1) (2022), 1–9.
  • S. G. Krein: Functional analysis, Noordhoff, Groningen (1972).
  • A. M. Perelomov: On the completeness of a system of coherent states, Theor. Math. Phys., 6 (1971), 156–164.
  • A. M. Sedletskii, Analytic Fourier transforms and exponential approximations. I., J. Math. Sci., 129 (6) (2005), 4251–4408.
  • O. V. Shavala: On some approximation properties of the Bessel functions of order $-5/2$, Mat. Stud., 43 (2) (2015), 180–184. (in Ukrainian)
  • O. V. Shavala: On completeness of systems of functions generated by the Bessel function, Bukovinian Math. J., 5 (3-4) (2017), 168–171. (in Ukrainian)
  • A. A. Shkalikov: Boundary problems for ordinary differential equations with parameter in the boundary conditions, J. Math. Sci., 33 (6) (1986), 1311–1342.
  • K. Stempak: On convergence and divergence of Fourier-Bessel series, Electron. Trans. Numer. Anal., 14 (2002), 223–235.
  • C. Tretter: Linear operator pencils $A-\lambda B$ with discrete spectrum, Integr. Equ. Oper. Theory., 37 (3) (2000), 357–373.
  • V. S. Vladimirov: Equations of mathematical physics, Nauka, Moscow (1981). (in Russian)
  • B. V. Vynnyts’kyi, V. M. Dilnyi: On approximation properties of one trigonometric system, Russ. Math., 58 (11) (2014), 10–21.
  • B. V. Vynnyts’kyi, R. V. Khats’: Some approximation properties of the systems of Bessel functions of index $-3/2$, Mat. Stud., 34 (2) (2010), 152–159.
  • B. V. Vynnyts’kyi, R. V. Khats’: Completeness and minimality of systems of Bessel functions, Ufa Math. J., 5 (2) (2013), 131–141.
  • B. V. Vynnyts’kyi, R. V. Khats’: On the completeness and minimality of sets of Bessel functions in weighted $L^2$-spaces, Eurasian Math. J., 6 (1) (2015), 123–131.
  • B. V. Vynnyts’kyi, R. V. Khats’: A remark on basis property of systems of Bessel and Mittag-Leffler type functions, J. Contemp. Math. Anal. (Armen. Acad. Sci.), 50 (6) (2015), 300–305.
  • B. V. Vynnyts’kyi, R. V. Khats’: Complete biorthogonal systems of Bessel functions, Mat. Stud., 48 (2) (2017), 150–155.
  • B. V. Vynnyts’kyi, R. V. Khats’ and I. B. Sheparovych: Unconditional bases of systems of Bessel functions, Eurasian Math. J., 11 (4) (2020), 76–86.
  • B. V. Vynnyts’kyi, O. V. Shavala: Boundedness of solutions of a second-order linear differential equation and a boundary value problem for Bessel’s equation, Mat. Stud., 30 (1) (2008), 31–41. (in Ukrainian).
  • B. V. Vynnyts’kyi, O. V. Shavala: Some properties of boundary value problems for Bessel’s equation, Math. Bull. Shevchenko Sci. Soc., 10 (2013), 169–172.
  • G. N. Watson: A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge (1944).
Yıl 2022, Cilt: 5 Sayı: 2, 60 - 71, 15.06.2022
https://doi.org/10.33205/cma.1077842

Öz

Kaynakça

  • H. Bateman, A. Erdélyi: Higher transcendental functions, Vol. 2, McGraw-Hill, New York-Toronto-London (1953).
  • Yu. M. Berezanskii: Expansions in eigenfunctions of self-adjoint operators, Transl. Math. Monogr., Vol. 17. Amer. Math. Soc., Providence, RI (1968).
  • M. Bertola, M. Gekhtman and J. Szmigielski: Cubic string boundary value problems and Cauchy biorthogonal polynomials, J. Phys. A: Math. Theor., 42 (45) (2009), 13 p.
  • C. Cesarano: A note on bi-orthogonal polynomials and functions, Fluids, 5 (3) (2020), 105.
  • N. Dunford, J. T. Schwartz: Linear operators. Spectral operators, Part III. Wiley-Interscience, New York-London-Sydney-Toronto (1971).
  • R. V. Khats’: On conditions of the completeness of some systems of Bessel functions in the space $L^2((0;1);x^{2p} dx)$, Azerb. J. Math., 11 (1) (2021), 3–10.
  • R. V. Khats’: Completeness conditions of systems of Bessel functions in weighted $L^2$-spaces in terms of entire functions, Turk. J. Math., 45 (2) (2021), 890–895.
  • R. V. Khats’: Integral representation of one class of entire functions, Armen. J. Math., 14 (1) (2022), 1–9.
  • S. G. Krein: Functional analysis, Noordhoff, Groningen (1972).
  • A. M. Perelomov: On the completeness of a system of coherent states, Theor. Math. Phys., 6 (1971), 156–164.
  • A. M. Sedletskii, Analytic Fourier transforms and exponential approximations. I., J. Math. Sci., 129 (6) (2005), 4251–4408.
  • O. V. Shavala: On some approximation properties of the Bessel functions of order $-5/2$, Mat. Stud., 43 (2) (2015), 180–184. (in Ukrainian)
  • O. V. Shavala: On completeness of systems of functions generated by the Bessel function, Bukovinian Math. J., 5 (3-4) (2017), 168–171. (in Ukrainian)
  • A. A. Shkalikov: Boundary problems for ordinary differential equations with parameter in the boundary conditions, J. Math. Sci., 33 (6) (1986), 1311–1342.
  • K. Stempak: On convergence and divergence of Fourier-Bessel series, Electron. Trans. Numer. Anal., 14 (2002), 223–235.
  • C. Tretter: Linear operator pencils $A-\lambda B$ with discrete spectrum, Integr. Equ. Oper. Theory., 37 (3) (2000), 357–373.
  • V. S. Vladimirov: Equations of mathematical physics, Nauka, Moscow (1981). (in Russian)
  • B. V. Vynnyts’kyi, V. M. Dilnyi: On approximation properties of one trigonometric system, Russ. Math., 58 (11) (2014), 10–21.
  • B. V. Vynnyts’kyi, R. V. Khats’: Some approximation properties of the systems of Bessel functions of index $-3/2$, Mat. Stud., 34 (2) (2010), 152–159.
  • B. V. Vynnyts’kyi, R. V. Khats’: Completeness and minimality of systems of Bessel functions, Ufa Math. J., 5 (2) (2013), 131–141.
  • B. V. Vynnyts’kyi, R. V. Khats’: On the completeness and minimality of sets of Bessel functions in weighted $L^2$-spaces, Eurasian Math. J., 6 (1) (2015), 123–131.
  • B. V. Vynnyts’kyi, R. V. Khats’: A remark on basis property of systems of Bessel and Mittag-Leffler type functions, J. Contemp. Math. Anal. (Armen. Acad. Sci.), 50 (6) (2015), 300–305.
  • B. V. Vynnyts’kyi, R. V. Khats’: Complete biorthogonal systems of Bessel functions, Mat. Stud., 48 (2) (2017), 150–155.
  • B. V. Vynnyts’kyi, R. V. Khats’ and I. B. Sheparovych: Unconditional bases of systems of Bessel functions, Eurasian Math. J., 11 (4) (2020), 76–86.
  • B. V. Vynnyts’kyi, O. V. Shavala: Boundedness of solutions of a second-order linear differential equation and a boundary value problem for Bessel’s equation, Mat. Stud., 30 (1) (2008), 31–41. (in Ukrainian).
  • B. V. Vynnyts’kyi, O. V. Shavala: Some properties of boundary value problems for Bessel’s equation, Math. Bull. Shevchenko Sci. Soc., 10 (2013), 169–172.
  • G. N. Watson: A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge (1944).
Toplam 27 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik
Bölüm Makaleler
Yazarlar

Ruslan Khats' 0000-0001-9905-5447

Yayımlanma Tarihi 15 Haziran 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 5 Sayı: 2

Kaynak Göster

APA Khats’, R. (2022). Generalized eigenvectors of linear operators and biorthogonal systems. Constructive Mathematical Analysis, 5(2), 60-71. https://doi.org/10.33205/cma.1077842
AMA Khats’ R. Generalized eigenvectors of linear operators and biorthogonal systems. CMA. Haziran 2022;5(2):60-71. doi:10.33205/cma.1077842
Chicago Khats’, Ruslan. “Generalized Eigenvectors of Linear Operators and Biorthogonal Systems”. Constructive Mathematical Analysis 5, sy. 2 (Haziran 2022): 60-71. https://doi.org/10.33205/cma.1077842.
EndNote Khats’ R (01 Haziran 2022) Generalized eigenvectors of linear operators and biorthogonal systems. Constructive Mathematical Analysis 5 2 60–71.
IEEE R. Khats’, “Generalized eigenvectors of linear operators and biorthogonal systems”, CMA, c. 5, sy. 2, ss. 60–71, 2022, doi: 10.33205/cma.1077842.
ISNAD Khats’, Ruslan. “Generalized Eigenvectors of Linear Operators and Biorthogonal Systems”. Constructive Mathematical Analysis 5/2 (Haziran 2022), 60-71. https://doi.org/10.33205/cma.1077842.
JAMA Khats’ R. Generalized eigenvectors of linear operators and biorthogonal systems. CMA. 2022;5:60–71.
MLA Khats’, Ruslan. “Generalized Eigenvectors of Linear Operators and Biorthogonal Systems”. Constructive Mathematical Analysis, c. 5, sy. 2, 2022, ss. 60-71, doi:10.33205/cma.1077842.
Vancouver Khats’ R. Generalized eigenvectors of linear operators and biorthogonal systems. CMA. 2022;5(2):60-71.