On Basic Fourier–Bessel Expansions

José Luis Cardoso

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

When dealing with Fourier expansions using the third Jackson (also known as Hahn–Exton) $q$-Bessel function, the corresponding positive zeros $j_{k\nu}$ and the “shifted” zeros, $qj_{k\nu}$, among others, play an essential role. Mixing classical analysis with $q$-analysis we were able to prove asymptotic relations between those zeros and the “shifted” ones, as well as the asymptotic behavior of the third Jackson $q$-Bessel function when computed on the “shifted” zeros. A version of a $q$-analogue of the Riemann–Lebesgue theorem within the scope of basic Fourier–Bessel expansions is also exhibited.

Détail

Keywords: third Jackson $q$-Bessel function; Hahn–Exton $q$-Bessel function; basic Fourier–Bessel expansions; basic hypergeometric function; asymptotic behavior; Riemann–Lebesgue theorem.

Bibliographie

[1] Abreu L. D., “Completeness, special functions and uncertainty principles over $q$-linear grids”, J. Phys. A: Math. Gen., 39 (2006), 14567–14580, arXiv: math.CA/0602440 | DOI | MR

[2] Abreu L. D., “Functions $q$-orthogonal with respect to their own zeros”, Proc. Amer. Math. Soc., 134 (2006), 2695–2701 | DOI | MR

[3] Abreu L. D., Bustoz J., “On the completeness of sets of $q$-Bessel functions $J_\nu^{(3)}(x;q)$”, Theory and Applications of Special Functions, Dev. Math., 13, Springer, New York, 2005, 29–38 | MR

[4] Abreu L. D., Bustoz J., Cardoso J. L., “The roots of the third Jackson $q$-Bessel function”, Int. J. Math. Math. Sci., 2003, 4241–4248 | DOI | MR

[5] Andrews G. E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999 | MR

[6] Annaby M. H., “$q$-type sampling theorems”, Results Math., 44 (2003), 214–225 | DOI | MR

[7] Annaby M. H., Mansour Z. S., “On the zeros of the second and third Jackson $q$-Bessel functions and their associated $q$-Hankel transforms”, Math. Proc. Cambridge Philos. Soc., 147 (2009), 47–67 | DOI | MR

[8] Bettaibi N., Bouzeffour F., Ben Elmonser H., Binous W., “Elements of harmonic analysis related to the third basic zero order Bessel function”, J. Math. Anal. Appl., 342 (2008), 1203–1219 | DOI | MR

[9] Bustoz J., Cardoso J. L., “Basic analog of Fourier series on a $q$-linear grid”, J. Approx. Theory, 112 (2001), 134–157 | DOI | MR

[10] Bustoz J., Suslov S. K., “Basic analog of Fourier series on a $q$-quadratic grid”, Methods Appl. Anal., 5 (1998), 1–38, arXiv: math.CA/9706216 | MR

[11] Cardoso J. L., “Basic {F}ourier series in a $q$-linear grid: convergence theorems”, J. Math. Anal. Appl., 323 (2006), 313–330 | DOI | MR

[12] Cardoso J. L., “Basic Fourier series: convergence on and outside the $q$-linear grid”, J. Fourier Anal. Appl., 17 (2011), 96–114, arXiv: math.CA/0605764 | DOI | MR

[13] Cardoso J. L., “A few properties of the third Jackson $q$-Bessel function”, Anal. Math., 42 (2016), 323–337 | DOI | MR

[14] Cardoso J. L., Petronilho J., “Variations around Jackson's quantum operator”, Methods Appl. Anal., 22 (2015), 343–358 | MR

[15] Cieśliński J. L., “Improved $q$-exponential and $q$-trigonometric functions”, Appl. Math. Lett., 24 (2011), 2110–2114, arXiv: 1006.5652 | DOI | MR

[16] Elmonser H., Sellami M., Fitouhi A., “Inequalities related to the third Jackson $q$-Bessel function of order zero”, J. Inequal. Appl., 2013 (2013), 289, 22 pp. | DOI | MR

[17] Exton H., “A basic analogue of the Bessel–Clifford equation”, Jñānābha, 8 (1978), 49–56 | MR

[18] Exton H., $q$-hypergeometric functions and applications, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press, New York, 1983 | MR

[19] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 35, Cambridge University Press, Cambridge, 1990 | MR

[20] Hardy G. H., “Notes on special systems of orthogonal functions (II): on functions orthogonal with respect to their own zeros”, J. London Math. Soc., 14 (1939), 37–44 | DOI | MR

[21] Hayman W. K., “On the zeros of a $q$-Bessel function”, Complex analysis and dynamical systems II, Contemp. Math., 382, Amer. Math. Soc., Providence, RI, 2005, 205–216 | DOI | MR

[22] Koelink H. T., Swarttouw R. F., “On the zeros of the Hahn–Exton $q$-Bessel function and associated $q$-Lommel polynomials”, J. Math. Anal. Appl., 186 (1994), 690–710 | DOI | MR

[23] Koornwinder T. H., Swarttouw R. F., “On $q$-analogues of the Fourier and Hankel transforms”, Trans. Amer. Math. Soc., 333 (1992), 445–461 | MR

[24] Olde Daalhuis A. B., “Asymptotic expansions for $q$-gamma, $q$-exponential, and $q$-Bessel functions”, J. Math. Anal. Appl., 186 (1994), 896–913 | DOI | MR

[25] Suslov S. K., ““{A}ddition” theorems for some $q$-exponential and $q$-trigonometric functions”, Methods Appl. Anal., 4 (1997), 11–32 | MR

[26] Suslov S. K., An introduction to basic Fourier series, Developments in Mathematics, 9, Kluwer Academic Publishers, Dordrecht, 2003 | DOI | MR

[27] Swarttouw R. F., The Hahn–Exton $q$-Bessel function, Ph.D. Thesis, Technische Universiteit Delft, 1992 | MR

[28] Štampach F., “Nevanlinna extremal measures for polynomials related to $q^{-1}$-Fibonacci polynomials”, Adv. in Appl. Math., 78 (2016), 56–75 | DOI | MR

[29] Štampach F., Šťovíček P., “The Nevanlinna parametrization for $q$-Lommel polynomials in the indeterminate case”, J. Approx. Theory, 201 (2016), 48–72, arXiv: 1407.0217 | DOI | MR

[30] Watson G. N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944 | MR

[31] Wilcox H. J., Myers D. L., An introduction to Lebesgue integration and Fourier series, Dover Publications, Inc., New York, 1994 | MR