Completeness and minimality of systems of Bessel functions

B. V. Vynnyts'kyi   ; R. V. Khats'

Ufa mathematical journal, Tome 5 (2013) no. 2, pp. 131-141

Résumé

We find the necessary and sufficient conditions for the completeness and minimality in the space $L^2(0;1)$ of system $(\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\Bbb N)$ generated by Bessel function of the first kind of index $\nu\ge -1/2$. Moreover, we establish a criterion for the completeness and minimality of system $(x^{-2}\sqrt{x\rho_k}J_{3/2}(x\rho_k):k\in\Bbb N)$ in the space $L^2((0;1);x^2 dx)$.

Détail

Keywords: Paley–Wiener theorem, Bessel function, entire function, complete system, minimal system, basis.
Mots-clés : biorthogonal system

Bibliographie

[1] Russian Mathematics (Izvestiya Vuzov. Matematika), 45:8 (2001), 1–7 | MR | Zbl

[2] N. I. Akhiezer, “To the theory of coupled integral equations”, Kharkov State University Press, 25 (1957), 5–31 (in Russian)

[3] H. Bateman, A. Erdélyi, Higher transcendental functions, v. 2, McGraw-Hill, New York–Toronto–London, 1953 | MR

[4] A. Benedek, R. Panzone, “Mean convergence of series of Bessel functions”, Rev. Un. Mat. Argentina, 26 (1972), 42–61 | MR | Zbl

[5] A. Benedek, R. Panzone, “On mean convergence of Fourier–Bessel series of negative order”, Stud. Appl. Math., 50 (1971), 281–292 | MR | Zbl

[6] A. Benedek, R. Panzone, “Pointwise convergence of series of Bessel functions”, Rev. Un. Mat. Argentina, 26 (1972), 167–186 | MR | Zbl

[7] J. J. Betancor, K. Stempak, “Relating multipliers and transplantation for Fourier–Bessel expansions and Hankel transform”, Tohoku Math. J., 53 (2001), 109–129 | DOI | MR | Zbl

[8] R. P. Boas, H. Pollard, “Complete sets of Bessel and Legendre functions”, Ann. of Math., 48:2 (1947), 366–384 | DOI | MR | Zbl

[9] Ó. Ciaurri, K. Stempak, J. L. Varona, “Mean Cesàro-type summability of Fourier–Neumann series”, Studia Sci. Math. Hungar., 42 (2005), 413–430 | MR | Zbl

[10] L. Colzani, A. Crespi, G. Travaglini, M. Vignati, “Equiconvergence theorems for Fourier–Bessel expansions with applications to the harmonic analysis of radial functions in euclidean and non euclidean spaces”, Trans. Amer. Math. Soc., 338 (1993), 43–55 | MR | Zbl

[11] M. M. Dzhrbashyan, Integral transforms and representations of functions in the complex domain, Nauka, M., 1966 (in Russian) | MR

[12] John B. Garnett, Bounded analytic functions, Academic Press, New York, 1981 | MR

[13] J. L. Griffith, “Hankel transforms of functions zero outside a finite interval”, J. Proc. Roy. Soc. New South Wales, 89 (1955), 109–115 | MR

[14] J. J. Guadalupe, M. Pérez, F. J. Ruiz, “Mean and weak convergence of Fourier–Bessel series”, J. Math. Anal. Appl., 173 (1993), 370–389 | DOI | MR | Zbl

[15] J. J. Guadalupe, M. Pérez, F. J. Ruiz, J. L. Varona, “Two notes on convergence and divergence a.e. of Fourier series with respect to some orthogonal systems”, Proc. Amer. Math. Soc., 116 (1992), 457–464 | MR | Zbl

[16] J. R. Higgins, Completeness and basis properties of sets of special functions, Cambridge University Press, Cambridge–London–New York–Melbourne, 1977 | MR | Zbl

[17] K. P. Isaev, R. S. Yulmukhametov, “Unconditional exponential bases in Hilbert spaces”, Ufa Math. J., 3:1 (2011), 3–15 | Zbl

[18] B. N. Khabibullin, Completeness of exponential systems and uniqueness sets, Bashkir State University Press, Ufa, 2008 (in Russian)

[19] P. Koosis, Introduction to $H_p$ spaces, Cambridge University Press, Cambridge, 1998 | MR

[20] B. Ya. Levin, Lectures on entire functions, Transl. Math. Monogr., 150, Amer. Math. Soc., Providence, RI, 1996 | MR | Zbl

[21] A. M. Sedletskii, “Analytic Fourier transforms and exponential approximations. I; II”, J. Math. Sci., 129:6 (2005), 4251–4408 | DOI | MR

[22] I. Singer, Bases in Banach Spaces, v. 1, Springer-Verlag, Berlin, 1970 | MR

[23] K. Stempak, “On convergence and divergence of Fourier–Bessel series”, Electron. Trans. Numer. Anal., 14 (2002), 223–235 | MR | Zbl

[24] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Second edition, Clarendon Press, Oxford, 1948

[25] Marcel Dekker, Inc., New York, 1971 | MR | Zbl

[26] B. V. Vynnyts'kyi, V. M. Dilnyi, “On some analogues of Paley–Wiener theorem and one boundary value problem for Bessel operator”, Proc. of Int. Conf. on complex analysis in memory of A. A. Gol'dberg (Lviv, Ukraine, May 31–June 5, 2010), 63–64

[27] 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 | MR | Zbl

[28] B. V. Vynnyts'kyi, R. V. Khats', “Approximation properties of Bessel functions and a generalization of the notion of eigenvector”, Proc. of Int. Conf. on complex analysis in memory of A. A. Gol'dberg (Lviv, Ukraine, May 31–June 5, 2010), 64–65

[29] B. V. Vynnyts'kyi, R. V. Khats', “A generalization of the notion of eigenvector”, Modern problems of analysis, Proc. of Int. Conf. (Chernivtsi, Ukraine, September 30–October 3, 2010), 51–52 (in Ukrainian)

[30] 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) | MR

[31] B. V. Vynnyts'kyi, O. V. Shavala, “On completeness of the system $(\cos(\rho_nx)+\rho_nx\sin(\rho_nx))$ and a boundary value problem for Bessel operator”, Analysis and topology, Proc. of Int. Conf. (Lviv, Ukraine, May 26–June 7, 2008), 54–55

[32] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944 | MR | Zbl

[33] G. M. Wing, “The mean convergence of orthogonal series”, Amer. J. Math., 72 (1950), 792–808 | DOI | MR