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On some properties of Buhmann functions

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Ukrainian Mathematical Journal Aims and scope

Abstract

We study functions introduced by Buhmann. The exact exponent of smoothness of these functions is obtained and the problem of positivity of their Hankel transforms is analyzed.

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References

  1. V. P. Zastavnyi, Positive-Definite Functions Depending on the Norm. Solution of the Schoenberg Problem [in Russian], Preprint No. 09.91, Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk (1991).

    Google Scholar 

  2. V. P. Zastavnyi, “On positive definiteness of some functions,” J. Multivar. Anal., 73, 55–81 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  3. R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer, Boston (2004).

    MATH  Google Scholar 

  4. E. M. Stein and G. Weiss, Introduction to Fourier Analysis of Euclidean Spaces, Princeton University, Princeton (1971).

    Google Scholar 

  5. H. Wendland, “Error estimates for interpolation by compactly supported radial basis functions of minimal degree,” J. Approxim. Theor., 93, 258–272 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  6. M. D. Buhmann, “A new class of radial basis functions with compact support,” Math. Comput., 70, No. 233, 307–318 (2000).

    MathSciNet  Google Scholar 

  7. V. P. Zastavnyi and R. M. Trigub, “Positive-definite splines of a special kind,” Mat. Sb., 193, No. 12, 41–68 (2002).

    MathSciNet  Google Scholar 

  8. V. P. Zastavnyi, “Positive-definite radial functions and splines,” Dokl. Ros. Akad. Nauk, 386, No. 4, 446–449 (2002).

    MathSciNet  Google Scholar 

  9. H. Wendland, “Piecewise polynomial, positive definite, and compactly supported radial functions of minimal degree,” Adv. Comput. Math., 4, 389–396 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  10. V. P. Zastavnyi and R. M. Trigub, Positive-Definite Splines [in Russian], Deposited at UkrNIINTI, No. 593-Uk.87, Donetsk (1987).

  11. R. M. Trigub, “Some properties of the Fourier transform of a measure and their applications,” in: Proc. of the Internat. Conf. on the Theory of Approximation of Functions (Kiev, 1983) [in Russian], Nauka, Moscow (1987), pp. 439–443.

    Google Scholar 

  12. R. M. Trigub, “Criterion of a characteristic function and the Pólya-type property for radial functions of several variables,” Teor. Ver. Primen., 34, 805–810 (1989).

    MathSciNet  MATH  Google Scholar 

  13. R. M. Trigub, “Positive-definite radial functions and splines,” in: Abstr. of the Internat. Conf. on the Constructive Theory of Functions (Varna, May 1987) [in Russian], Sofia (1987), p. 123.

  14. R. M. Trigub, “Positive-definite functions and splines,” in: Proc. of the Fifth Saratov Winter School on the Theory of Functions and Approximation’ (25.01–04.02.1990) [in Russian], Part 1, Saratov (1992), pp. 68–75.

  15. R. M. Trigub, Some Topics in Fourier Analysis and Approximation Theory, Preprint, Internet: funct-an/9612008, Donetsk (1995).

  16. R. M. Trigub, “On positive definite radial splines of a special form,” in: Abstr. of the Internat. Conf. OFEA’2001 on the Optimization of Finite-Element Approximation & Splines and Wavelets,” St. Petersburg (2001), p. 174.

  17. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University, Cambridge (1945).

    Google Scholar 

  18. N. I. Akhiezer, Lectures on Integral Transformations [in Russian], Vyshcha Shkola, Kharkov (1984).

    MATH  Google Scholar 

  19. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford (1937).

    MATH  Google Scholar 

  20. R. Askey and H. Pollard, “Some absolutely monotonic and completely monotonic functions,” SIAM J. Math. Anal., 5, 58–63 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  21. J. L. Fields and M. E. Ismail, “On the positivity of some 1 F′2 s,” SIAM J. Math. Anal., 6, 551–559 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Askey and G. Gasper, “Positive Jacobi polynomial sums, II,” Amer. J. Math., 98, No. 3, 709–737 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Makai, “On a monotonic property of certain Sturm-Liouville functions,” Acta Math. Acad. Sci. Hung., 3, 165–172 (1952).

    Article  MathSciNet  MATH  Google Scholar 

  24. D. S. Moak, “Completely monotonic functions of the form s −b(s 2 + 1)−a,” Rocky Mountain J. Math., 17, No. 4, 719–725 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Steinig, “The sign of Lommel’s function,” Trans. Amer. Math. Soc., 163, 123–129 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Gasper, “Positive integrals of Bessel functions,” SIAM J. Math. Anal., 6, No. 5, 868–881 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  27. R. G. Cooke, “A monotonic property of Bessel functions,” Math. Comput., 12, 180–185 (1937).

    MATH  Google Scholar 

  28. T. Gneiting, K. Konis, and D. Richards, “Experimental approaches to Kuttner’s problem,” Exp. Math., 10, No. 1, 117–124 (2001).

    MathSciNet  MATH  Google Scholar 

  29. O. Yu. Pasenchenko, “Sufficient conditions for the characteristic function of a two-dimensional isotropic distribution,” Teor. Imov. Mat. Statist., Issue 53, 138–141 (1995).

  30. B. Kuttner, “On the Riesz means of a Fourier series,” J. London Math. Soc., 19, No. 2, 77–84 (1944).

    MathSciNet  MATH  Google Scholar 

  31. B. I. Golubov, “On Abel-Poisson-type and Riesz means,” Anal. Math., 7, 161–184 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  32. J. K. Misiewicz and D. S. P. Richards, “Positivity of integrals of Bessel functions,” SIAM J. Math. Anal., 25, No. 2, 596–601 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  33. P. K. Suetin, Classical Orthogonal Polynomials [in Russia], Nauka, Moscow (1979).

    MATH  Google Scholar 

  34. V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus [in Russian], Fizmatgiz, Moscow (1974).

    Google Scholar 

  35. E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” J. London Math. Soc., 10, 286–293 (1935).

    MATH  Google Scholar 

  36. E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” Proc. London Math. Soc., 46, No. 2, 389–408 (1940).

    MathSciNet  MATH  Google Scholar 

  37. B. Kuttner, “On positive Riesz and Abel typical means,” Proc. London Math. Soc., 49, 328–352 (1947).

    MathSciNet  MATH  Google Scholar 

  38. V. P. Zastavnyi, “Sufficient conditions for the integrability of Hankel transforms,” Tr. Inst. Prikl. Mat. Mekh. NAN Ukr., Issue 9, 68–75 (2004).

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Authors and Affiliations

  1. Donetsk National University, Donetsk

    V. P. Zastavnyi

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  1. V. P. Zastavnyi
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__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1045–1067, August, 2006.

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Zastavnyi, V.P. On some properties of Buhmann functions. Ukr Math J 58, 1184–1208 (2006). https://doi.org/10.1007/s11253-006-0128-z

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  • DOI: https://doi.org/10.1007/s11253-006-0128-z

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