An extension of positivity for integrals of Bessel functions and Buhmann’s radial basis functions

Cho, Yong-Kum; Chung, Seok-Young; Yun, Hera

doi:10.1090/bproc/34

An extension of positivity for integrals of Bessel functions and Buhmann’s radial basis functions
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by Yong-Kum Cho, Seok-Young Chung and Hera Yun HTML | PDF

Proc. Amer. Math. Soc. Ser. B 5 (2018), 25-39

Abstract:

As to the Bessel integrals of type \begin{equation*} \int _0^x \left (x^\mu -t^\mu \right )^\lambda t^\alpha J_\beta (t)dt\qquad (x>0), \end{equation*} we improve known positivity results by making use of new positivity criteria for ${}_1F_2$ and ${}_2F_3$ generalized hypergeometric functions. As an application, we extend Buhmann’s class of compactly supported radial basis functions.

References

Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145, DOI 10.1137/1.9781611970470
Richard Askey, Problems which interest and/or annoy me, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, pp. 3–15. MR 1246848, DOI 10.1016/0377-0427(93)90312-Y
W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. MR 0185155
M. D. Buhmann, A new class of radial basis functions with compact support, Math. Comp. 70 (2001), no. 233, 307–318. MR 1803129, DOI 10.1090/S0025-5718-00-01251-5
Y.-K. Cho and H. Yun, Newton diagram of positivity for ${}_1F_2$ generalized hypergeometric functions, Preprint, arXiv:1801.02312 (2018)
Jerry L. Fields and Mourad El-Houssieny Ismail, On the positivity of some $_{1}F_{2}$’s, SIAM J. Math. Anal. 6 (1975), 551–559. MR 361189, DOI 10.1137/0506050
George Gasper, Positive integrals of Bessel functions, SIAM J. Math. Anal. 6 (1975), no. 5, 868–881. MR 390318, DOI 10.1137/0506076
George Gasper, Positive sums of the classical orthogonal polynomials, SIAM J. Math. Anal. 8 (1977), no. 3, 423–447. MR 432946, DOI 10.1137/0508032
Tilmann Gneiting, Radial positive definite functions generated by Euclid’s hat, J. Multivariate Anal. 69 (1999), no. 1, 88–119. MR 1701408, DOI 10.1006/jmva.1998.1800
Tilmann Gneiting, Kjell Konis, and Donald Richards, Experimental approaches to Kuttner’s problem, Experiment. Math. 10 (2001), no. 1, 117–124. MR 1822857, DOI 10.1080/10586458.2001.10504434
B. I. Golubov, On Abel-Poisson type and Riesz means, Anal. Math. 7 (1981), no. 3, 161–184 (English, with Russian summary). MR 635483, DOI 10.1007/BF01908520
B. Kuttner, On the Riesz means of a Fourier series. II, J. London Math. Soc. 19 (1944), 77–84. MR 12686, DOI 10.1112/jlms/19.74_{P}art_{2}.77
Yudell L. Luke, The special functions and their approximations, Vol. I, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR 0241700
Jolanta K. Misiewicz and Donald St. P. Richards, Positivity of integrals of Bessel functions, SIAM J. Math. Anal. 25 (1994), no. 2, 596–601. MR 1266579, DOI 10.1137/S0036141092226934
Robert Schaback, The missing Wendland functions, Adv. Comput. Math. 34 (2011), no. 1, 67–81. MR 2783302, DOI 10.1007/s10444-009-9142-7
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
Holger Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), no. 4, 389–396. MR 1366510, DOI 10.1007/BF02123482
Holger Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR 2131724
R. E. Williamson, Multiply monotone functions and their Laplace transforms, Duke Math. J. 23 (1956), 189–207. MR 77581, DOI 10.1215/S0012-7094-56-02317-1
V. P. Zastavnyĭ, On some properties of the Buhmann functions, Ukraïn. Mat. Zh. 58 (2006), no. 8, 1045–1067 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 58 (2006), no. 8, 1184–1208. MR 2345078, DOI 10.1007/s11253-006-0128-z