Advanced Search
Article Contents
Article Contents

General types of spherical mean operators and $K$-functionals of fractional orders

Abstract Related Papers Cited by
  • We design a general type of spherical mean operators and employ them to approximate $L_p$ class functions. We show that optimal orders of approximation are achieved via appropriately defined K-functionals of fractional orders.
    Mathematics Subject Classification: Primary: 40A05, 41A36; Secondary: 41A60, 45C05, 47A75.


    \begin{equation} \\ \end{equation}
  • [1]

    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series 55, U.S. Government Printing Office, Washington, D.C., 1964.


    G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications, No. 71. Cambridge University Press, Cambridge, 1999.doi: 10.1017/CBO9781107325937.


    W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964.


    E. Belinsky, F. Dai and Z. Ditzian, Multivariate approximating averages, J. Approx. Theory 125, 1 (2003), 85-105.doi: 10.1016/j.jat.2003.09.005.


    W. O. Bray, Growth and integrability of Fourier transforms on Euclidean spaces, to appear in J. Fourier Analysis and Applications, arXiv:1308.2268. doi: 10.1007/s00041-014-9354-1.


    W. O. Bray and M. A. Pinsky, Growth properties of Fourier transforms via moduli of continuity, J. Funct. Anal. 255, 9 (2008), 2265-2285.doi: 10.1016/j.jfa.2008.06.017.


    D. Chen, V. A. Menegatto and X. Sun, A necessary and sufficient condition for strictly positive definite functions on spheres, Proceedings of the American Mathematical Society 131, 9 (2003), 2733-2740.doi: 10.1090/S0002-9939-03-06730-3.


    F. Dai and Z. Ditzian, Combinations of multivariate averages, J. Approx. Theory 131, 2 (2004), 268-283.doi: 10.1016/j.jat.2004.10.003.


    Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar. 81, 4 (1998), 323-348.doi: 10.1023/A:1006554907440.


    Z. Ditzian, Smoothness of a function and the growth of its Fourier transform or its Fourier coefficients, J. Approx. Theory 162, 5 (2010), 980-986.doi: 10.1016/j.jat.2009.11.003.


    Z. Ditzian, Relating smoothness to expressions involving Fourier coefficient or to a Fourier transform, J. Approx. Theory 164, 10 (2012), 1369-1389.doi: 10.1016/j.jat.2012.05.004.


    R. E. Edwards, Fourier Series: A Modern Introduction, Vol. II, Holt, Rinehart and Winston, INC., New York, 1967.


    D. Gioev, Moduli of continuity and average decay of Fourier transforms: two sided estimates, in Integrable Systems and Random Matrices (Cont. Math. 458), Amer. Math. Soc., Providence, RI, (2008), 377-392.doi: 10.1090/conm/458/08948.


    D. Gorbachev and S. Tikhonov, Moduli of smoothness and growth properties of Fourier transforms: two sided estimates, J. Approx. Theory 164, 9 (2012), 1283-1312.doi: 10.1016/j.jat.2012.05.017.


    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6nd edition, translated from the Russian sixth edition, translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, Academic Press, Inc., San Diego, CA, 2000.


    F. J. Narcowich, X. Sun and J. D. Ward, Approximation power of RBFs and their associated SBFs: a connection, Adv. Comput. Math. 27, 1 (2007), 107-124.doi: 10.1007/s10444-005-7506-1.


    I. J. Schoenberg, Metric spaces and completely monotone functions, The Annals of Mathematics 39, 4 (1938), 811-841.doi: 10.2307/1968466.


    I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J., 9 (1988), 96-108.


    J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994.doi: 10.1007/978-1-4612-0873-0.


    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.


    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971.


    E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 3nd edition, Chelsea Publishing Co., New York, 1986.


    C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems - B, 4 (2004), 1065-1089.doi: 10.3934/dcdsb.2004.4.1065.

  • 加载中

Article Metrics

HTML views() PDF downloads(94) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint