Complete biorthogonal systems of Bessel functions

B. V. Vynnyts’kyi; R. V. Khats’

doi:10.15330/ms.48.1.150-155

Let

$\nu\geq-1/2$ and

$(\rho_k)_{k\in\mathbb N}$ be a sequence of nonzero complex numbers such that

$\rho_k^2\neq \rho_m^2$ for

$k\neq m$ . We prove that if the system

$\big\{\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\mathbb N\big\}$ of Bessel functions of the first kind of index

$\nu\geq-1/2$ is exact (i.e. complete and minimal) in the space

$L^2(0;1)$ , then its biorthogonal system is also exact in

$L^2(0;1)$ .

1. N.I. Akhiezer, To the theory of paired integral equations, Uchenye Zapiski Kharkov. Gos. Univ., 25 (1957), 5–31 (in Russian).

2. R.P. Boas, H. Pollard, Complete sets of Bessel and Legendre functions, Ann. of Math., 48 (1947), №2, 366–384.

3. M.M. Dzhrbashyan, Integral transforms and representations of functions in the complex domain, Nauka, Moscow, 1966 (in Russian).

4. J.L. Griffith, Hankel transforms of functions zero outside a finite interval, J. Proc. Roy. Soc. New South Wales, 89 (1955), 109–115.

5. H. Hochstadt, The mean convergence of Fourier-Bessel series, SIAM Rev., 9 (1967), 211–218.

6. B.Ya. Levin, Lectures on entire functions, Transl. Math. Monographs, 150, Amer. Math. Soc., Providence, R.I., 1996.

7. K. Stempak, On convergence and divergence of Fourier-Bessel series, Electron. Trans. Numer. Anal., 14 (2002), 223–235.

8. V.S. Vladimirov, Equations of mathematical physics, Nauka, Moscow, 1981 (in Russian). Engl. transl.: Mir Publishers, Moscow, 1984.

9. B.V. Vynnyts’kyi, R.V. Khats’, Completeness and minimality of systems of Bessel functions, Ufa Math. J., 5 (2013), №2, 131–141.

10. B.V. Vynnyts’kyi, R.V. Khats’, A remark on basis property of systems of Bessel and Mittag-Leffler type functions, Izv. NAN Armenii, Matematika, 50 (2015), №6, 16–25 (in Russian). Engl. transl.: J. Contemp. Math. Anal., 50 (2015), №6, 300–305.

11. 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 (2015), №1, 123–131.

12. B.V. Vynnyts’kyi, R.V. Khats’, Some approximation properties of the systems of Bessel functions of index

$-3/2$ , Mat. Stud., 34 (2010), №2, 152–159.

13. G.N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944.

14. R.M. Young, An introduction to nonharmonic Fourier series, Pure and Appl. Math., 93, Academic Press, New York, 1980.

15. R.M. Young, On complete biorthogonal systems, Proc. Amer. Math. Soc., 83 (1981), №3, 537–540.

	Complete biorthogonal systems of Bessel functions
Author	B. V. Vynnyts’kyi, R. V. Khats’ vynnytskyi@ukr.net; khats@ukr.net Institute of Physics, Mathematics, Economy and Innovation Technologies, Drohobych Ivan Franko State Pedagogical University, Drohobych, Ukraine
Abstract	Let $\nu\geq-1/2$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of nonzero complex numbers such that $\rho_k^2\neq \rho_m^2$ for $k\neq m$ . We prove that if the system $\big\{\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\mathbb N\big\}$ of Bessel functions of the first kind of index $\nu\geq-1/2$ is exact (i.e. complete and minimal) in the space $L^2(0;1)$ , then its biorthogonal system is also exact in $L^2(0;1)$ .
Keywords	Bessel function; entire function of exponential type; complete system; minimal system; biorthogonal system; exact system; orthonormal basis
DOI	doi:10.15330/ms.48.2.150-155
Reference	1. N.I. Akhiezer, To the theory of paired integral equations, Uchenye Zapiski Kharkov. Gos. Univ., 25 (1957), 5–31 (in Russian). 2. R.P. Boas, H. Pollard, Complete sets of Bessel and Legendre functions, Ann. of Math., 48 (1947), №2, 366–384. 3. M.M. Dzhrbashyan, Integral transforms and representations of functions in the complex domain, Nauka, Moscow, 1966 (in Russian). 4. J.L. Griffith, Hankel transforms of functions zero outside a finite interval, J. Proc. Roy. Soc. New South Wales, 89 (1955), 109–115. 5. H. Hochstadt, The mean convergence of Fourier-Bessel series, SIAM Rev., 9 (1967), 211–218. 6. B.Ya. Levin, Lectures on entire functions, Transl. Math. Monographs, 150, Amer. Math. Soc., Providence, R.I., 1996. 7. K. Stempak, On convergence and divergence of Fourier-Bessel series, Electron. Trans. Numer. Anal., 14 (2002), 223–235. 8. V.S. Vladimirov, Equations of mathematical physics, Nauka, Moscow, 1981 (in Russian). Engl. transl.: Mir Publishers, Moscow, 1984. 9. B.V. Vynnyts’kyi, R.V. Khats’, Completeness and minimality of systems of Bessel functions, Ufa Math. J., 5 (2013), №2, 131–141. 10. B.V. Vynnyts’kyi, R.V. Khats’, A remark on basis property of systems of Bessel and Mittag-Leffler type functions, Izv. NAN Armenii, Matematika, 50 (2015), №6, 16–25 (in Russian). Engl. transl.: J. Contemp. Math. Anal., 50 (2015), №6, 300–305. 11. 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 (2015), №1, 123–131. 12. B.V. Vynnyts’kyi, R.V. Khats’, Some approximation properties of the systems of Bessel functions of index $-3/2$ , Mat. Stud., 34 (2010), №2, 152–159. 13. G.N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944. 14. R.M. Young, An introduction to nonharmonic Fourier series, Pure and Appl. Math., 93, Academic Press, New York, 1980. 15. R.M. Young, On complete biorthogonal systems, Proc. Amer. Math. Soc., 83 (1981), №3, 537–540.
Pages	150-155
Volume	48
Issue	2
Year	2017
Journal	Matematychni Studii
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